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section \<open>Examples for the \<open>real_asymp\<close> method\<close>
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theory Real_Asymp_Examples
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imports Real_Asymp
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begin
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context
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includes asymp_equiv_notation
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begin
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subsection \<open>Dominik Gruntz's examples\<close>
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lemma "((\<lambda>x::real. exp x * (exp (1/x - exp (-x)) - exp (1/x))) \<longlongrightarrow> -1) at_top"
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by real_asymp
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lemma "((\<lambda>x::real. exp x * (exp (1/x + exp (-x) + exp (-(x^2))) -
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exp (1/x - exp (-exp x)))) \<longlongrightarrow> 1) at_top"
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by real_asymp
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lemma "filterlim (\<lambda>x::real. exp (exp (x - exp (-x)) / (1 - 1/x)) - exp (exp x)) at_top at_top"
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by real_asymp
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text \<open>This example is notable because it produces an expansion of level 9.\<close>
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lemma "filterlim (\<lambda>x::real. exp (exp (exp x / (1 - 1/x))) -
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exp (exp (exp x / (1 - 1/x - ln x powr (-ln x))))) at_bot at_top"
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by real_asymp
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lemma "filterlim (\<lambda>x::real. exp (exp (exp (x + exp (-x)))) / exp (exp (exp x))) at_top at_top"
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by real_asymp
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lemma "filterlim (\<lambda>x::real. exp (exp (exp x)) / (exp (exp (exp (x - exp (-exp x)))))) at_top at_top"
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by real_asymp
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lemma "((\<lambda>x::real. exp (exp (exp x)) / exp (exp (exp x - exp (-exp (exp x))))) \<longlongrightarrow> 1) at_top"
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by real_asymp
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lemma "((\<lambda>x::real. exp (exp x) / exp (exp x - exp (-exp (exp x)))) \<longlongrightarrow> 1) at_top"
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by real_asymp
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lemma "filterlim (\<lambda>x::real. ln x ^ 2 * exp (sqrt (ln x) * ln (ln x) ^ 2 *
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exp (sqrt (ln (ln x)) * ln (ln (ln x)) ^ 3)) / sqrt x) (at_right 0) at_top"
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by real_asymp
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lemma "((\<lambda>x::real. (x * ln x * ln (x * exp x - x^2) ^ 2) /
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ln (ln (x^2 + 2*exp (exp (3*x^3*ln x))))) \<longlongrightarrow> 1/3) at_top"
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by real_asymp
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lemma "((\<lambda>x::real. (exp (x * exp (-x) / (exp (-x) + exp (-(2*x^2)/(x+1)))) - exp x) / x)
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\<longlongrightarrow> -exp 2) at_top"
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by real_asymp
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lemma "((\<lambda>x::real. (3 powr x + 5 powr x) powr (1/x)) \<longlongrightarrow> 5) at_top"
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by real_asymp
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lemma "filterlim (\<lambda>x::real. x / (ln (x powr (ln x powr (ln 2 / ln x))))) at_top at_top"
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by real_asymp
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lemma "filterlim (\<lambda>x::real. exp (exp (2*ln (x^5 + x) * ln (ln x))) /
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exp (exp (10*ln x * ln (ln x)))) at_top at_top"
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by real_asymp
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lemma "filterlim (\<lambda>x::real. 4/9 * (exp (exp (5/2*x powr (-5/7) + 21/8*x powr (6/11) +
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2*x powr (-8) + 54/17*x powr (49/45))) ^ 8) / (ln (ln (-ln (4/3*x powr (-5/14))))))
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at_top at_top"
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by real_asymp
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lemma "((\<lambda>x::real. (exp (4*x*exp (-x) /
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(1/exp x + 1/exp (2*x^2/(x+1)))) - exp x) / ((exp x)^4)) \<longlongrightarrow> 1) at_top "
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by real_asymp
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lemma "((\<lambda>x::real. exp (x*exp (-x) / (exp (-x) + exp (-2*x^2/(x+1))))/exp x) \<longlongrightarrow> 1) at_top"
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by real_asymp
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lemma "((\<lambda>x::real. exp (exp (-x/(1+exp (-x)))) * exp (-x/(1+exp (-x/(1+exp (-x))))) *
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exp (exp (-x+exp (-x/(1+exp (-x))))) / (exp (-x/(1+exp (-x))))^2 - exp x + x)
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\<longlongrightarrow> 2) at_top"
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by real_asymp
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lemma "((\<lambda>x::real. (ln(ln x + ln (ln x)) - ln (ln x)) /
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(ln (ln x + ln (ln (ln x)))) * ln x) \<longlongrightarrow> 1) at_top"
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by real_asymp
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lemma "((\<lambda>x::real. exp (ln (ln (x + exp (ln x * ln (ln x)))) /
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(ln (ln (ln (exp x + x + ln x)))))) \<longlongrightarrow> exp 1) at_top"
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by real_asymp
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lemma "((\<lambda>x::real. exp x * (sin (1/x + exp (-x)) - sin (1/x + exp (-(x^2))))) \<longlongrightarrow> 1) at_top"
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by real_asymp
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lemma "((\<lambda>x::real. exp (exp x) * (exp (sin (1/x + exp (-exp x))) - exp (sin (1/x)))) \<longlongrightarrow> 1) at_top"
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by real_asymp
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lemma "((\<lambda>x::real. max x (exp x) / ln (min (exp (-x)) (exp (-exp x)))) \<longlongrightarrow> -1) at_top"
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by real_asymp
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text \<open>The following example is taken from the paper by Richardson \<^emph>\<open>et al\<close>.\<close>
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lemma
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defines "f \<equiv> (\<lambda>x::real. ln (ln (x * exp (x * exp x) + 1)) - exp (exp (ln (ln x) + 1 / x)))"
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shows "(f \<longlongrightarrow> 0) at_top" (is ?thesis1)
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"f \<sim> (\<lambda>x. -(ln x ^ 2) / (2*x))" (is ?thesis2)
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unfolding f_def by real_asymp+
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subsection \<open>Asymptotic inequalities related to the Akra--Bazzi theorem\<close>
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definition "akra_bazzi_asymptotic1 b hb e p x \<longleftrightarrow>
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(1 - hb * inverse b * ln x powr -(1+e)) powr p * (1 + ln (b*x + hb*x/ln x powr (1+e)) powr (-e/2))
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\<ge> 1 + (ln x powr (-e/2) :: real)"
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definition "akra_bazzi_asymptotic1' b hb e p x \<longleftrightarrow>
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(1 + hb * inverse b * ln x powr -(1+e)) powr p * (1 + ln (b*x + hb*x/ln x powr (1+e)) powr (-e/2))
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\<ge> 1 + (ln x powr (-e/2) :: real)"
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definition "akra_bazzi_asymptotic2 b hb e p x \<longleftrightarrow>
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(1 + hb * inverse b * ln x powr -(1+e)) powr p * (1 - ln (b*x + hb*x/ln x powr (1+e)) powr (-e/2))
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\<le> 1 - ln x powr (-e/2 :: real)"
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definition "akra_bazzi_asymptotic2' b hb e p x \<longleftrightarrow>
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(1 - hb * inverse b * ln x powr -(1+e)) powr p * (1 - ln (b*x + hb*x/ln x powr (1+e)) powr (-e/2))
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\<le> 1 - ln x powr (-e/2 :: real)"
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definition "akra_bazzi_asymptotic3 e x \<longleftrightarrow> (1 + (ln x powr (-e/2))) / 2 \<le> (1::real)"
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definition "akra_bazzi_asymptotic4 e x \<longleftrightarrow> (1 - (ln x powr (-e/2))) * 2 \<ge> (1::real)"
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definition "akra_bazzi_asymptotic5 b hb e x \<longleftrightarrow>
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ln (b*x - hb*x*ln x powr -(1+e)) powr (-e/2::real) < 1"
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definition "akra_bazzi_asymptotic6 b hb e x \<longleftrightarrow> hb / ln x powr (1 + e :: real) < b/2"
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definition "akra_bazzi_asymptotic7 b hb e x \<longleftrightarrow> hb / ln x powr (1 + e :: real) < (1 - b) / 2"
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definition "akra_bazzi_asymptotic8 b hb e x \<longleftrightarrow> x*(1 - b - hb / ln x powr (1 + e :: real)) > 1"
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definition "akra_bazzi_asymptotics b hb e p x \<longleftrightarrow>
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akra_bazzi_asymptotic1 b hb e p x \<and> akra_bazzi_asymptotic1' b hb e p x \<and>
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akra_bazzi_asymptotic2 b hb e p x \<and> akra_bazzi_asymptotic2' b hb e p x \<and>
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akra_bazzi_asymptotic3 e x \<and> akra_bazzi_asymptotic4 e x \<and> akra_bazzi_asymptotic5 b hb e x \<and>
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akra_bazzi_asymptotic6 b hb e x \<and> akra_bazzi_asymptotic7 b hb e x \<and>
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akra_bazzi_asymptotic8 b hb e x"
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lemmas akra_bazzi_asymptotic_defs =
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akra_bazzi_asymptotic1_def akra_bazzi_asymptotic1'_def
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akra_bazzi_asymptotic2_def akra_bazzi_asymptotic2'_def akra_bazzi_asymptotic3_def
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akra_bazzi_asymptotic4_def akra_bazzi_asymptotic5_def akra_bazzi_asymptotic6_def
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akra_bazzi_asymptotic7_def akra_bazzi_asymptotic8_def akra_bazzi_asymptotics_def
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lemma akra_bazzi_asymptotics:
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assumes "\<And>b. b \<in> set bs \<Longrightarrow> b \<in> {0<..<1}" and "e > 0"
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shows "eventually (\<lambda>x. \<forall>b\<in>set bs. akra_bazzi_asymptotics b hb e p x) at_top"
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proof (intro eventually_ball_finite ballI)
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fix b assume "b \<in> set bs"
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with assms have "b \<in> {0<..<1}" by simp
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with \<open>e > 0\<close> show "eventually (\<lambda>x. akra_bazzi_asymptotics b hb e p x) at_top"
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unfolding akra_bazzi_asymptotic_defs
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by (intro eventually_conj; real_asymp simp: mult_neg_pos)
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qed simp
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lemma
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fixes b \<epsilon> :: real
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assumes "b \<in> {0<..<1}" and "\<epsilon> > 0"
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shows "filterlim (\<lambda>x. (1 - H / (b * ln x powr (1 + \<epsilon>))) powr p *
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(1 + ln (b * x + H * x / ln x powr (1+\<epsilon>)) powr (-\<epsilon>/2)) -
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(1 + ln x powr (-\<epsilon>/2))) (at_right 0) at_top"
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using assms by (real_asymp simp: mult_neg_pos)
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context
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fixes b e p :: real
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assumes assms: "b > 0" "e > 0"
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begin
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lemmas [simp] = mult_neg_pos
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real_limit "(\<lambda>x. ((1 - 1 / (b * ln x powr (1 + e))) powr p *
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(1 + ln (b * x + x / ln x powr (1+e)) powr (-e/2)) -
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(1 + ln x powr (-e/2))) * ln x powr ((e / 2) + 1))"
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facts: assms
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end
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context
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fixes b \<epsilon> :: real
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assumes assms: "b > 0" "\<epsilon> > 0" "\<epsilon> < 1 / 4"
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begin
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real_expansion "(\<lambda>x. (1 - H / (b * ln x powr (1 + \<epsilon>))) powr p *
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(1 + ln (b * x + H * x / ln x powr (1+\<epsilon>)) powr (-\<epsilon>/2)) -
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(1 + ln x powr (-\<epsilon>/2)))"
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terms: 10 (strict)
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facts: assms
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end
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context
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fixes e :: real
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assumes e: "e > 0" "e < 1 / 4"
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begin
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real_expansion "(\<lambda>x. (1 - 1 / (1/2 * ln x powr (1 + e))) *
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(1 + ln (1/2 * x + x / ln x powr (1+e)) powr (-e/2)) -
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(1 + ln x powr (-e/2)))"
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terms: 10 (strict)
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facts: e
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end
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subsection \<open>Concrete Mathematics\<close>
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text \<open>The following inequalities are discussed on p.\ 441 in Concrete Mathematics.\<close>
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lemma
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fixes c \<epsilon> :: real
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assumes "0 < \<epsilon>" "\<epsilon> < 1" "1 < c"
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shows "(\<lambda>_::real. 1) \<in> o(\<lambda>x. ln (ln x))"
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and "(\<lambda>x::real. ln (ln x)) \<in> o(\<lambda>x. ln x)"
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and "(\<lambda>x::real. ln x) \<in> o(\<lambda>x. x powr \<epsilon>)"
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and "(\<lambda>x::real. x powr \<epsilon>) \<in> o(\<lambda>x. x powr c)"
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and "(\<lambda>x. x powr c) \<in> o(\<lambda>x. x powr ln x)"
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and "(\<lambda>x. x powr ln x) \<in> o(\<lambda>x. c powr x)"
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and "(\<lambda>x. c powr x) \<in> o(\<lambda>x. x powr x)"
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and "(\<lambda>x. x powr x) \<in> o(\<lambda>x. c powr (c powr x))"
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using assms by (real_asymp (verbose))+
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subsection \<open>Stack Exchange\<close>
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text \<open>
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http://archives.math.utk.edu/visual.calculus/1/limits.15/
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\<close>
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lemma "filterlim (\<lambda>x::real. (x * sin x) / abs x) (at_right 0) (at 0)"
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by real_asymp
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lemma "filterlim (\<lambda>x::real. x^2 / (sqrt (x^2 + 12) - sqrt (12))) (nhds (12 / sqrt 3)) (at 0)"
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proof -
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note [simp] = powr_half_sqrt sqrt_def (* TODO: Better simproc for sqrt/root? *)
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have "sqrt (12 :: real) = sqrt (4 * 3)"
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by simp
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also have "\<dots> = 2 * sqrt 3" by (subst real_sqrt_mult) simp
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finally show ?thesis by real_asymp
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qed
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text \<open>\<^url>\<open>http://math.stackexchange.com/questions/625574\<close>\<close>
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lemma "(\<lambda>x::real. (1 - 1/2 * x^2 - cos (x / (1 - x^2))) / x^4) \<midarrow>0\<rightarrow> 23/24"
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by real_asymp
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text \<open>\<^url>\<open>http://math.stackexchange.com/questions/122967\<close>\<close>
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real_limit "\<lambda>x. (x + 1) powr (1 + 1 / x) - x powr (1 + 1 / (x + a))"
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lemma "((\<lambda>x::real. ((x + 1) powr (1 + 1/x) - x powr (1 + 1 / (x + a)))) \<longlongrightarrow> 1) at_top"
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by real_asymp
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real_limit "\<lambda>x. x ^ 2 * (arctan x - pi / 2) + x"
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lemma "filterlim (\<lambda>x::real. x^2 * (arctan x - pi/2) + x) (at_right 0) at_top"
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by real_asymp
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real_limit "\<lambda>x. (root 3 (x ^ 3 + x ^ 2 + x + 1) - sqrt (x ^ 2 + x + 1) * ln (exp x + x) / x)"
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lemma "((\<lambda>x::real. root 3 (x^3 + x^2 + x + 1) - sqrt (x^2 + x + 1) * ln (exp x + x) / x)
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\<longlongrightarrow> -1/6) at_top"
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by real_asymp
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context
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fixes a b :: real
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assumes ab: "a > 0" "b > 0"
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begin
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real_limit "\<lambda>x. ((a powr x - x * ln a) / (b powr x - x * ln b)) powr (1 / x ^ 2)"
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limit: "at_right 0" facts: ab
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real_limit "\<lambda>x. ((a powr x - x * ln a) / (b powr x - x * ln b)) powr (1 / x ^ 2)"
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limit: "at_left 0" facts: ab
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lemma "(\<lambda>x. ((a powr x - x * ln a) / (b powr x - x * ln b)) powr (1 / x ^ 2))
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\<midarrow>0\<rightarrow> exp (ln a * ln a / 2 - ln b * ln b / 2)" using ab by real_asymp
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end
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text \<open>\<^url>\<open>http://math.stackexchange.com/questions/547538\<close>\<close>
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lemma "(\<lambda>x::real. ((x+4) powr (3/2) + exp x - 9) / x) \<midarrow>0\<rightarrow> 4"
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by real_asymp
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text \<open>\<^url>\<open>https://www.freemathhelp.com/forum/threads/93513\<close>\<close>
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lemma "((\<lambda>x::real. ((3 powr x + 4 powr x) / 4) powr (1/x)) \<longlongrightarrow> 4) at_top"
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by real_asymp
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lemma "((\<lambda>x::real. x powr (3/2) * (sqrt (x + 1) + sqrt (x - 1) - 2 * sqrt x)) \<longlongrightarrow> -1/4) at_top"
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by real_asymp
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text \<open>\<^url>\<open>https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/limcondirectory/LimitConstant.html\<close>\<close>
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lemma "(\<lambda>x::real. (cos (2*x) - 1) / (cos x - 1)) \<midarrow>0\<rightarrow> 4"
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by real_asymp
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lemma "(\<lambda>x::real. tan (2*x) / (x - pi/2)) \<midarrow>pi/2\<rightarrow> 2"
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by real_asymp
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text \<open>@url{"https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/liminfdirectory/LimitInfinity.html"}\<close>
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lemma "filterlim (\<lambda>x::real. (3 powr x + 3 powr (2*x)) powr (1/x)) (nhds 9) at_top"
|
|
295 |
using powr_def[of 3 "2::real"] by real_asymp
|
|
296 |
|
69597
|
297 |
text \<open>\<^url>\<open>https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/lhopitaldirectory/LHopital.html\<close>\<close>
|
68630
|
298 |
lemma "filterlim (\<lambda>x::real. (x^2 - 1) / (x^2 + 3*x - 4)) (nhds (2/5)) (at 1)"
|
|
299 |
by real_asymp
|
|
300 |
|
|
301 |
lemma "filterlim (\<lambda>x::real. (x - 4) / (sqrt x - 2)) (nhds 4) (at 4)"
|
|
302 |
by real_asymp
|
|
303 |
|
|
304 |
lemma "filterlim (\<lambda>x::real. sin x / x) (at_left 1) (at 0)"
|
|
305 |
by real_asymp
|
|
306 |
|
|
307 |
lemma "filterlim (\<lambda>x::real. (3 powr x - 2 powr x) / (x^2 - x)) (nhds (ln (2/3))) (at 0)"
|
|
308 |
by (real_asymp simp: ln_div)
|
|
309 |
|
|
310 |
lemma "filterlim (\<lambda>x::real. (1/x - 1/3) / (x^2 - 9)) (nhds (-1/54)) (at 3)"
|
|
311 |
by real_asymp
|
|
312 |
|
|
313 |
lemma "filterlim (\<lambda>x::real. (x * tan x) / sin (3*x)) (nhds 0) (at 0)"
|
|
314 |
by real_asymp
|
|
315 |
|
|
316 |
lemma "filterlim (\<lambda>x::real. sin (x^2) / (x * tan x)) (at 1) (at 0)"
|
|
317 |
by real_asymp
|
|
318 |
|
|
319 |
lemma "filterlim (\<lambda>x::real. (x^2 * exp x) / tan x ^ 2) (at 1) (at 0)"
|
|
320 |
by real_asymp
|
|
321 |
|
|
322 |
lemma "filterlim (\<lambda>x::real. exp (-1/x^2) / x^2) (at 0) (at 0)"
|
|
323 |
by real_asymp
|
|
324 |
|
|
325 |
lemma "filterlim (\<lambda>x::real. exp x / (5 * x + 200)) at_top at_top"
|
|
326 |
by real_asymp
|
|
327 |
|
|
328 |
lemma "filterlim (\<lambda>x::real. (3 + ln x) / (x^2 + 7)) (at 0) at_top"
|
|
329 |
by real_asymp
|
|
330 |
|
|
331 |
lemma "filterlim (\<lambda>x::real. (x^2 + 3*x - 10) / (7*x^2 - 5*x + 4)) (at_right (1/7)) at_top"
|
|
332 |
by real_asymp
|
|
333 |
|
|
334 |
lemma "filterlim (\<lambda>x::real. (ln x ^ 2) / exp (2*x)) (at_right 0) at_top"
|
|
335 |
by real_asymp
|
|
336 |
|
|
337 |
lemma "filterlim (\<lambda>x::real. (3 * x + 2 powr x) / (2 * x + 3 powr x)) (at 0) at_top"
|
|
338 |
by real_asymp
|
|
339 |
|
|
340 |
lemma "filterlim (\<lambda>x::real. (exp x + 2 / x) / (exp x + 5 / x)) (at_left 1) at_top"
|
|
341 |
by real_asymp
|
|
342 |
|
|
343 |
lemma "filterlim (\<lambda>x::real. sqrt (x^2 + 1) - sqrt (x + 1)) at_top at_top"
|
|
344 |
by real_asymp
|
|
345 |
|
|
346 |
|
|
347 |
lemma "filterlim (\<lambda>x::real. x * ln x) (at_left 0) (at_right 0)"
|
|
348 |
by real_asymp
|
|
349 |
|
|
350 |
lemma "filterlim (\<lambda>x::real. x * ln x ^ 2) (at_right 0) (at_right 0)"
|
|
351 |
by real_asymp
|
|
352 |
|
|
353 |
lemma "filterlim (\<lambda>x::real. ln x * tan x) (at_left 0) (at_right 0)"
|
|
354 |
by real_asymp
|
|
355 |
|
|
356 |
lemma "filterlim (\<lambda>x::real. x powr sin x) (at_left 1) (at_right 0)"
|
|
357 |
by real_asymp
|
|
358 |
|
|
359 |
lemma "filterlim (\<lambda>x::real. (1 + 3 / x) powr x) (at_left (exp 3)) at_top"
|
|
360 |
by real_asymp
|
|
361 |
|
|
362 |
lemma "filterlim (\<lambda>x::real. (1 - x) powr (1 / x)) (at_left (exp (-1))) (at_right 0)"
|
|
363 |
by real_asymp
|
|
364 |
|
|
365 |
lemma "filterlim (\<lambda>x::real. (tan x) powr x^2) (at_left 1) (at_right 0)"
|
|
366 |
by real_asymp
|
|
367 |
|
|
368 |
lemma "filterlim (\<lambda>x::real. x powr (1 / sqrt x)) (at_right 1) at_top"
|
|
369 |
by real_asymp
|
|
370 |
|
|
371 |
lemma "filterlim (\<lambda>x::real. ln x powr (1/x)) (at_right 1) at_top"
|
|
372 |
by (real_asymp (verbose))
|
|
373 |
|
|
374 |
lemma "filterlim (\<lambda>x::real. x powr (x powr x)) (at_right 0) (at_right 0)"
|
|
375 |
by (real_asymp (verbose))
|
|
376 |
|
|
377 |
|
69597
|
378 |
text \<open>\<^url>\<open>http://calculus.nipissingu.ca/problems/limit_problems.html\<close>\<close>
|
68630
|
379 |
lemma "((\<lambda>x::real. (x^2 - 1) / \<bar>x - 1\<bar>) \<longlongrightarrow> -2) (at_left 1)"
|
|
380 |
"((\<lambda>x::real. (x^2 - 1) / \<bar>x - 1\<bar>) \<longlongrightarrow> 2) (at_right 1)"
|
|
381 |
by real_asymp+
|
|
382 |
|
|
383 |
lemma "((\<lambda>x::real. (root 3 x - 1) / \<bar>sqrt x - 1\<bar>) \<longlongrightarrow> -2 / 3) (at_left 1)"
|
|
384 |
"((\<lambda>x::real. (root 3 x - 1) / \<bar>sqrt x - 1\<bar>) \<longlongrightarrow> 2 / 3) (at_right 1)"
|
|
385 |
by real_asymp+
|
|
386 |
|
|
387 |
|
69597
|
388 |
text \<open>\<^url>\<open>https://math.stackexchange.com/questions/547538\<close>\<close>
|
68630
|
389 |
|
|
390 |
lemma "(\<lambda>x::real. ((x + 4) powr (3/2) + exp x - 9) / x) \<midarrow>0\<rightarrow> 4"
|
|
391 |
by real_asymp
|
|
392 |
|
69597
|
393 |
text \<open>\<^url>\<open>https://math.stackexchange.com/questions/625574\<close>\<close>
|
68630
|
394 |
lemma "(\<lambda>x::real. (1 - x^2 / 2 - cos (x / (1 - x^2))) / x^4) \<midarrow>0\<rightarrow> 23/24"
|
|
395 |
by real_asymp
|
|
396 |
|
69597
|
397 |
text \<open>\<^url>\<open>https://www.mapleprimes.com/questions/151308-A-Hard-Limit-Question\<close>\<close>
|
68630
|
398 |
lemma "(\<lambda>x::real. x / (x - 1) - 1 / ln x) \<midarrow>1\<rightarrow> 1 / 2"
|
|
399 |
by real_asymp
|
|
400 |
|
69597
|
401 |
text \<open>\<^url>\<open>https://www.freemathhelp.com/forum/threads/93513-two-extremely-difficult-limit-problems\<close>\<close>
|
68630
|
402 |
lemma "((\<lambda>x::real. ((3 powr x + 4 powr x) / 4) powr (1/x)) \<longlongrightarrow> 4) at_top"
|
|
403 |
by real_asymp
|
|
404 |
|
|
405 |
lemma "((\<lambda>x::real. x powr 1.5 * (sqrt (x + 1) + sqrt (x - 1) - 2 * sqrt x)) \<longlongrightarrow> -1/4) at_top"
|
|
406 |
by real_asymp
|
|
407 |
|
69597
|
408 |
text \<open>\<^url>\<open>https://math.stackexchange.com/questions/1390833\<close>\<close>
|
68630
|
409 |
context
|
|
410 |
fixes a b :: real
|
|
411 |
assumes ab: "a + b > 0" "a + b = 1"
|
|
412 |
begin
|
|
413 |
real_limit "\<lambda>x. (a * x powr (1 / x) + b) powr (x / ln x)"
|
|
414 |
facts: ab
|
|
415 |
end
|
|
416 |
|
|
417 |
|
|
418 |
subsection \<open>Unsorted examples\<close>
|
|
419 |
|
|
420 |
context
|
|
421 |
fixes a :: real
|
|
422 |
assumes a: "a > 1"
|
|
423 |
begin
|
|
424 |
|
|
425 |
text \<open>
|
|
426 |
It seems that Mathematica fails to expand the following example when \<open>a\<close> is a variable.
|
|
427 |
\<close>
|
|
428 |
lemma "(\<lambda>x. 1 - (1 - 1 / x powr a) powr x) \<sim> (\<lambda>x. x powr (1 - a))"
|
|
429 |
using a by real_asymp
|
|
430 |
|
|
431 |
real_limit "\<lambda>x. (1 - (1 - 1 / x powr a) powr x) * x powr (a - 1)"
|
|
432 |
facts: a
|
|
433 |
|
|
434 |
lemma "(\<lambda>n. log 2 (real ((n + 3) choose 3) / 4) + 1) \<sim> (\<lambda>n. 3 / ln 2 * ln n)"
|
|
435 |
proof -
|
|
436 |
have "(\<lambda>n. log 2 (real ((n + 3) choose 3) / 4) + 1) =
|
|
437 |
(\<lambda>n. log 2 ((real n + 1) * (real n + 2) * (real n + 3) / 24) + 1)"
|
|
438 |
by (subst binomial_gbinomial)
|
|
439 |
(simp add: gbinomial_pochhammer' numeral_3_eq_3 pochhammer_Suc add_ac)
|
|
440 |
moreover have "\<dots> \<sim> (\<lambda>n. 3 / ln 2 * ln n)"
|
|
441 |
by (real_asymp simp: divide_simps)
|
|
442 |
ultimately show ?thesis by simp
|
|
443 |
qed
|
|
444 |
|
|
445 |
end
|
|
446 |
|
|
447 |
lemma "(\<lambda>x::real. exp (sin x) / ln (cos x)) \<sim>[at 0] (\<lambda>x. -2 / x ^ 2)"
|
|
448 |
by real_asymp
|
|
449 |
|
|
450 |
real_limit "\<lambda>x. ln (1 + 1 / x) * x"
|
|
451 |
real_limit "\<lambda>x. ln x powr ln x / x"
|
|
452 |
real_limit "\<lambda>x. (arctan x - pi/2) * x"
|
|
453 |
real_limit "\<lambda>x. arctan (1/x) * x"
|
|
454 |
|
|
455 |
context
|
|
456 |
fixes c :: real assumes c: "c \<ge> 2"
|
|
457 |
begin
|
|
458 |
lemma c': "c^2 - 3 > 0"
|
|
459 |
proof -
|
|
460 |
from c have "c^2 \<ge> 2^2" by (rule power_mono) auto
|
|
461 |
thus ?thesis by simp
|
|
462 |
qed
|
|
463 |
|
|
464 |
real_limit "\<lambda>x. (c^2 - 3) * exp (-x)"
|
|
465 |
real_limit "\<lambda>x. (c^2 - 3) * exp (-x)" facts: c'
|
|
466 |
end
|
|
467 |
|
|
468 |
lemma "c < 0 \<Longrightarrow> ((\<lambda>x::real. exp (c*x)) \<longlongrightarrow> 0) at_top"
|
|
469 |
by real_asymp
|
|
470 |
|
|
471 |
lemma "filterlim (\<lambda>x::real. -exp (1/x)) (at_left 0) (at_left 0)"
|
|
472 |
by real_asymp
|
|
473 |
|
|
474 |
lemma "((\<lambda>t::real. t^2 / (exp t - 1)) \<longlongrightarrow> 0) at_top"
|
|
475 |
by real_asymp
|
|
476 |
|
|
477 |
lemma "(\<lambda>n. (1 + 1 / real n) ^ n) \<longlonglongrightarrow> exp 1"
|
|
478 |
by real_asymp
|
|
479 |
|
|
480 |
lemma "((\<lambda>x::real. (1 + y / x) powr x) \<longlongrightarrow> exp y) at_top"
|
|
481 |
by real_asymp
|
|
482 |
|
|
483 |
lemma "eventually (\<lambda>x::real. x < x^2) at_top"
|
|
484 |
by real_asymp
|
|
485 |
|
|
486 |
lemma "eventually (\<lambda>x::real. 1 / x^2 \<ge> 1 / x) (at_left 0)"
|
|
487 |
by real_asymp
|
|
488 |
|
|
489 |
lemma "A > 1 \<Longrightarrow> (\<lambda>n. ((1 + 1 / sqrt A) / 2) powr real_of_int (2 ^ Suc n)) \<longlonglongrightarrow> 0"
|
|
490 |
by (real_asymp simp: divide_simps add_pos_pos)
|
|
491 |
|
|
492 |
lemma
|
|
493 |
assumes "c > (1 :: real)" "k \<noteq> 0"
|
|
494 |
shows "(\<lambda>n. real n ^ k) \<in> o(\<lambda>n. c ^ n)"
|
|
495 |
using assms by (real_asymp (verbose))
|
|
496 |
|
|
497 |
lemma "(\<lambda>x::real. exp (-(x^4))) \<in> o(\<lambda>x. exp (-(x^4)) + 1 / x ^ n)"
|
|
498 |
by real_asymp
|
|
499 |
|
|
500 |
lemma "(\<lambda>x::real. x^2) \<in> o[at_right 0](\<lambda>x. x)"
|
|
501 |
by real_asymp
|
|
502 |
|
|
503 |
lemma "(\<lambda>x::real. x^2) \<in> o[at_left 0](\<lambda>x. x)"
|
|
504 |
by real_asymp
|
|
505 |
|
|
506 |
lemma "(\<lambda>x::real. 2 * x + x ^ 2) \<in> \<Theta>[at_left 0](\<lambda>x. x)"
|
|
507 |
by real_asymp
|
|
508 |
|
|
509 |
lemma "(\<lambda>x::real. inverse (x - 1)^2) \<in> \<omega>[at 1](\<lambda>x. x)"
|
|
510 |
by real_asymp
|
|
511 |
|
|
512 |
lemma "(\<lambda>x::real. sin (1 / x)) \<sim> (\<lambda>x::real. 1 / x)"
|
|
513 |
by real_asymp
|
|
514 |
|
|
515 |
lemma
|
|
516 |
assumes "q \<in> {0<..<1}"
|
|
517 |
shows "LIM x at_left 1. log q (1 - x) :> at_top"
|
|
518 |
using assms by real_asymp
|
|
519 |
|
|
520 |
context
|
|
521 |
fixes x k :: real
|
|
522 |
assumes xk: "x > 1" "k > 0"
|
|
523 |
begin
|
|
524 |
|
|
525 |
lemmas [simp] = add_pos_pos
|
|
526 |
|
|
527 |
real_expansion "\<lambda>l. sqrt (1 + l * (l + 1) * 4 * pi^2 * k^2 * (log x (l + 1) - log x l)^2)"
|
|
528 |
terms: 2 facts: xk
|
|
529 |
|
|
530 |
lemma
|
|
531 |
"(\<lambda>l. sqrt (1 + l * (l + 1) * 4 * pi^2 * k^2 * (log x (l + 1) - log x l)^2) -
|
|
532 |
sqrt (1 + 4 * pi\<^sup>2 * k\<^sup>2 / (ln x ^ 2))) \<in> O(\<lambda>l. 1 / l ^ 2)"
|
|
533 |
using xk by (real_asymp simp: inverse_eq_divide power_divide root_powr_inverse)
|
|
534 |
|
|
535 |
end
|
|
536 |
|
|
537 |
lemma "(\<lambda>x. (2 * x^3 - 128) / (sqrt x - 2)) \<midarrow>4\<rightarrow> 384"
|
|
538 |
by real_asymp
|
|
539 |
|
|
540 |
lemma
|
|
541 |
"((\<lambda>x::real. (x^2 - 1) / abs (x - 1)) \<longlongrightarrow> 2) (at_right 1)"
|
|
542 |
"((\<lambda>x::real. (x^2 - 1) / abs (x - 1)) \<longlongrightarrow> -2) (at_left 1)"
|
|
543 |
by real_asymp+
|
|
544 |
|
|
545 |
lemma "(\<lambda>x::real. (root 3 x - 1) / (sqrt x - 1)) \<midarrow>1\<rightarrow> 2/3"
|
|
546 |
by real_asymp
|
|
547 |
|
|
548 |
lemma
|
|
549 |
fixes a b :: real
|
|
550 |
assumes "a > 1" "b > 1" "a \<noteq> b"
|
|
551 |
shows "((\<lambda>x. ((a powr x - x * ln a) / (b powr x - x * ln b)) powr (1/x^3)) \<longlongrightarrow> 1) at_top"
|
|
552 |
using assms by real_asymp
|
|
553 |
|
|
554 |
|
|
555 |
context
|
|
556 |
fixes a b :: real
|
|
557 |
assumes ab: "a > 0" "b > 0"
|
|
558 |
begin
|
|
559 |
|
|
560 |
lemma
|
|
561 |
"((\<lambda>x. ((a powr x - x * ln a) / (b powr x - x * ln b)) powr (1 / x ^ 2)) \<longlongrightarrow>
|
|
562 |
exp ((ln a ^ 2 - ln b ^ 2) / 2)) (at 0)"
|
|
563 |
using ab by (real_asymp simp: power2_eq_square)
|
|
564 |
|
|
565 |
end
|
|
566 |
|
|
567 |
real_limit "\<lambda>x. 1 / sin (1/x) ^ 2 + 1 / tan (1/x) ^ 2 - 2 * x ^ 2"
|
|
568 |
|
|
569 |
real_limit "\<lambda>x. ((1 / x + 4) powr 1.5 + exp (1 / x) - 9) * x"
|
|
570 |
|
|
571 |
lemma "x > (1 :: real) \<Longrightarrow>
|
|
572 |
((\<lambda>n. abs (x powr n / (n * (1 + x powr (2 * n)))) powr (1 / n)) \<longlongrightarrow> 1 / x) at_top"
|
|
573 |
by (real_asymp simp add: exp_minus field_simps)
|
|
574 |
|
|
575 |
lemma "x = (1 :: real) \<Longrightarrow>
|
|
576 |
((\<lambda>n. abs (x powr n / (n * (1 + x powr (2 * n)))) powr (1 / n)) \<longlongrightarrow> 1 / x) at_top"
|
|
577 |
by (real_asymp simp add: exp_minus field_simps)
|
|
578 |
|
|
579 |
lemma "x > (0 :: real) \<Longrightarrow> x < 1 \<Longrightarrow>
|
|
580 |
((\<lambda>n. abs (x powr n / (n * (1 + x powr (2 * n)))) powr (1 / n)) \<longlongrightarrow> x) at_top"
|
|
581 |
by real_asymp
|
|
582 |
|
|
583 |
lemma "(\<lambda>x. (exp (sin b) - exp (sin (b * x))) * tan (pi * x / 2)) \<midarrow>1\<rightarrow>
|
|
584 |
(2*b/pi * exp (sin b) * cos b)"
|
|
585 |
by real_asymp
|
|
586 |
|
|
587 |
(* SLOW *)
|
|
588 |
lemma "filterlim (\<lambda>x::real. (sin (tan x) - tan (sin x))) (at 0) (at 0)"
|
|
589 |
by real_asymp
|
|
590 |
|
|
591 |
(* SLOW *)
|
|
592 |
lemma "(\<lambda>x::real. 1 / sin x powr (tan x ^ 2)) \<midarrow>pi/2\<rightarrow> exp (1 / 2)"
|
|
593 |
by (real_asymp simp: exp_minus)
|
|
594 |
|
|
595 |
lemma "a > 0 \<Longrightarrow> b > 0 \<Longrightarrow> c > 0 \<Longrightarrow>
|
|
596 |
filterlim (\<lambda>x::real. ((a powr x + b powr x + c powr x) / 3) powr (3 / x))
|
|
597 |
(nhds (a * b * c)) (at 0)"
|
|
598 |
by (real_asymp simp: exp_add add_divide_distrib exp_minus algebra_simps)
|
|
599 |
|
|
600 |
real_expansion "\<lambda>x. arctan (sin (1 / x))"
|
|
601 |
|
|
602 |
real_expansion "\<lambda>x. ln (1 + 1 / x)"
|
|
603 |
terms: 5 (strict)
|
|
604 |
|
|
605 |
real_expansion "\<lambda>x. sqrt (x + 1) - sqrt (x - 1)"
|
|
606 |
terms: 3 (strict)
|
|
607 |
|
|
608 |
|
|
609 |
text \<open>
|
|
610 |
In this example, the first 7 terms of \<open>tan (sin x)\<close> and \<open>sin (tan x)\<close> coincide, which makes
|
|
611 |
the calculation a bit slow.
|
|
612 |
\<close>
|
|
613 |
real_limit "\<lambda>x. (tan (sin x) - sin (tan x)) / x ^ 7" limit: "at_right 0"
|
|
614 |
|
|
615 |
(* SLOW *)
|
|
616 |
real_expansion "\<lambda>x. sin (tan (1/x)) - tan (sin (1/x))"
|
|
617 |
terms: 1 (strict)
|
|
618 |
|
|
619 |
(* SLOW *)
|
|
620 |
real_expansion "\<lambda>x. tan (1 / x)"
|
|
621 |
terms: 6
|
|
622 |
|
|
623 |
real_expansion "\<lambda>x::real. arctan x"
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|
624 |
|
|
625 |
real_expansion "\<lambda>x. sin (tan (1/x))"
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626 |
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|
627 |
real_expansion "\<lambda>x. (sin (-1 / x) ^ 2) powr sin (-1/x)"
|
|
628 |
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|
629 |
real_limit "\<lambda>x. exp (max (sin x) 1)"
|
|
630 |
|
|
631 |
lemma "filterlim (\<lambda>x::real. 1 - 1 / x + ln x) at_top at_top"
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|
632 |
by (force intro: tendsto_diff filterlim_tendsto_add_at_top
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|
633 |
real_tendsto_divide_at_top filterlim_ident ln_at_top)
|
|
634 |
|
|
635 |
lemma "filterlim (\<lambda>i::real. i powr (-i/(i-1)) - i powr (-1/(i-1))) (at_left 1) (at_right 0)"
|
|
636 |
by real_asymp
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|
637 |
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|
638 |
lemma "filterlim (\<lambda>i::real. i powr (-i/(i-1)) - i powr (-1/(i-1))) (at_right (-1)) at_top"
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|
639 |
by real_asymp
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|
640 |
|
|
641 |
lemma "eventually (\<lambda>i::real. i powr (-i/(i-1)) - i powr (-1/(i-1)) < 1) (at_right 0)"
|
|
642 |
and "eventually (\<lambda>i::real. i powr (-i/(i-1)) - i powr (-1/(i-1)) > -1) at_top"
|
|
643 |
by real_asymp+
|
|
644 |
|
|
645 |
end
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|
646 |
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|
647 |
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|
648 |
subsection \<open>Interval arithmetic tests\<close>
|
|
649 |
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|
650 |
lemma "filterlim (\<lambda>x::real. x + sin x) at_top at_top"
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|
651 |
"filterlim (\<lambda>x::real. sin x + x) at_top at_top"
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|
652 |
"filterlim (\<lambda>x::real. x + (sin x + sin x)) at_top at_top"
|
|
653 |
by real_asymp+
|
|
654 |
|
|
655 |
lemma "filterlim (\<lambda>x::real. 2 * x + sin x * x) at_infinity at_top"
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|
656 |
"filterlim (\<lambda>x::real. 2 * x + (sin x + 1) * x) at_infinity at_top"
|
|
657 |
"filterlim (\<lambda>x::real. 3 * x + (sin x - 1) * x) at_infinity at_top"
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|
658 |
"filterlim (\<lambda>x::real. 2 * x + x * sin x) at_infinity at_top"
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|
659 |
"filterlim (\<lambda>x::real. 2 * x + x * (sin x + 1)) at_infinity at_top"
|
|
660 |
"filterlim (\<lambda>x::real. 3 * x + x * (sin x - 1)) at_infinity at_top"
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|
661 |
|
|
662 |
"filterlim (\<lambda>x::real. x + sin x * sin x) at_infinity at_top"
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|
663 |
"filterlim (\<lambda>x::real. x + sin x * (sin x + 1)) at_infinity at_top"
|
|
664 |
"filterlim (\<lambda>x::real. x + sin x * (sin x - 1)) at_infinity at_top"
|
|
665 |
"filterlim (\<lambda>x::real. x + sin x * (sin x + 2)) at_infinity at_top"
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|
666 |
"filterlim (\<lambda>x::real. x + sin x * (sin x - 2)) at_infinity at_top"
|
|
667 |
|
|
668 |
"filterlim (\<lambda>x::real. x + (sin x - 1) * sin x) at_infinity at_top"
|
|
669 |
"filterlim (\<lambda>x::real. x + (sin x - 1) * (sin x + 1)) at_infinity at_top"
|
|
670 |
"filterlim (\<lambda>x::real. x + (sin x - 1) * (sin x - 1)) at_infinity at_top"
|
|
671 |
"filterlim (\<lambda>x::real. x + (sin x - 1) * (sin x + 2)) at_infinity at_top"
|
|
672 |
"filterlim (\<lambda>x::real. x + (sin x - 1) * (sin x - 2)) at_infinity at_top"
|
|
673 |
|
|
674 |
"filterlim (\<lambda>x::real. x + (sin x - 2) * sin x) at_infinity at_top"
|
|
675 |
"filterlim (\<lambda>x::real. x + (sin x - 2) * (sin x + 1)) at_infinity at_top"
|
|
676 |
"filterlim (\<lambda>x::real. x + (sin x - 2) * (sin x - 1)) at_infinity at_top"
|
|
677 |
"filterlim (\<lambda>x::real. x + (sin x - 2) * (sin x + 2)) at_infinity at_top"
|
|
678 |
"filterlim (\<lambda>x::real. x + (sin x - 2) * (sin x - 2)) at_infinity at_top"
|
|
679 |
|
|
680 |
"filterlim (\<lambda>x::real. x + (sin x + 1) * sin x) at_infinity at_top"
|
|
681 |
"filterlim (\<lambda>x::real. x + (sin x + 1) * (sin x + 1)) at_infinity at_top"
|
|
682 |
"filterlim (\<lambda>x::real. x + (sin x + 1) * (sin x - 1)) at_infinity at_top"
|
|
683 |
"filterlim (\<lambda>x::real. x + (sin x + 1) * (sin x + 2)) at_infinity at_top"
|
|
684 |
"filterlim (\<lambda>x::real. x + (sin x + 1) * (sin x - 2)) at_infinity at_top"
|
|
685 |
|
|
686 |
"filterlim (\<lambda>x::real. x + (sin x + 2) * sin x) at_infinity at_top"
|
|
687 |
"filterlim (\<lambda>x::real. x + (sin x + 2) * (sin x + 1)) at_infinity at_top"
|
|
688 |
"filterlim (\<lambda>x::real. x + (sin x + 2) * (sin x - 1)) at_infinity at_top"
|
|
689 |
"filterlim (\<lambda>x::real. x + (sin x + 2) * (sin x + 2)) at_infinity at_top"
|
|
690 |
"filterlim (\<lambda>x::real. x + (sin x + 2) * (sin x - 2)) at_infinity at_top"
|
|
691 |
by real_asymp+
|
|
692 |
|
|
693 |
lemma "filterlim (\<lambda>x::real. x * inverse (sin x + 2)) at_top at_top"
|
|
694 |
"filterlim (\<lambda>x::real. x * inverse (sin x - 2)) at_bot at_top"
|
|
695 |
"filterlim (\<lambda>x::real. x / inverse (sin x + 2)) at_top at_top"
|
|
696 |
"filterlim (\<lambda>x::real. x / inverse (sin x - 2)) at_bot at_top"
|
|
697 |
by real_asymp+
|
|
698 |
|
|
699 |
lemma "filterlim (\<lambda>x::real. \<lfloor>x\<rfloor>) at_top at_top"
|
|
700 |
"filterlim (\<lambda>x::real. \<lceil>x\<rceil>) at_top at_top"
|
|
701 |
"filterlim (\<lambda>x::real. nat \<lfloor>x\<rfloor>) at_top at_top"
|
|
702 |
"filterlim (\<lambda>x::real. nat \<lceil>x\<rceil>) at_top at_top"
|
|
703 |
"filterlim (\<lambda>x::int. nat x) at_top at_top"
|
|
704 |
"filterlim (\<lambda>x::int. nat x) at_top at_top"
|
|
705 |
"filterlim (\<lambda>n::nat. real (n mod 42) + real n) at_top at_top"
|
|
706 |
by real_asymp+
|
|
707 |
|
|
708 |
lemma "(\<lambda>n. real_of_int \<lfloor>n\<rfloor>) \<sim>[at_top] (\<lambda>n. real_of_int n)"
|
|
709 |
"(\<lambda>n. sqrt \<lfloor>n\<rfloor>) \<sim>[at_top] (\<lambda>n. sqrt n)"
|
|
710 |
by real_asymp+
|
|
711 |
|
|
712 |
lemma
|
|
713 |
"c > 1 \<Longrightarrow> (\<lambda>n::nat. 2 ^ (n div c) :: int) \<in> o(\<lambda>n. 2 ^ n)"
|
|
714 |
by (real_asymp simp: field_simps)
|
|
715 |
|
|
716 |
lemma
|
|
717 |
"c > 1 \<Longrightarrow> (\<lambda>n::nat. 2 ^ (n div c) :: nat) \<in> o(\<lambda>n. 2 ^ n)"
|
|
718 |
by (real_asymp simp: field_simps)
|
|
719 |
|
|
720 |
end |