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Prove using definition of a limit that $\lim_{x\to \infty}\frac{\sin x}{x^2} = 0$.

Proof:

Let $\epsilon > 0$. Note that $\left|\frac{\sin x}{x^2}\right| \leq \frac 1 {x^2}$ for $x\ne 0$. Then choose $M= \frac 1{\sqrt{\epsilon}}$. Then if $x> M$ implies that $\left|\frac{\sin x}{x^2}\right| \leq \frac 1 {x^2} < \epsilon$.

Am I allowed to give an $M$ the way I did?

user10101
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1 Answers1

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Yes. Your proof is correct. That's how one chooses $M$. Though "Let we have for $x>M$" looks strange for me. The writing can be slightly modified as:

Let $\epsilon>0$. Note that $\displaystyle |\frac{\sin x}{x^2}|\le \frac{1}{x^2}$ for $x\ne 0$. Let $M=\frac{1}{\sqrt{\epsilon}}$. Then for $x>M$, one has $$ |\frac{\sin x}{x^2}|\le\frac{1}{x^2}<\epsilon\;. $$ Q.E.D.

  • Yes, typo. Thanks! – user10101 Nov 01 '21 at 02:36
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    @user10101 If the issue has been resolved, accepting and/or upvoting answers is the best way to say "thanks!": it scores points, signals resolution, and prevents bumping and automatic deletion. Two small suggestions for further editing: 1) delete "for"; 2) change the second "let" to "choose/put". – ryang Nov 01 '21 at 04:15
  • @ryang, yes. Was planning to accept answer. I am not new to this. There is a 5 min window before accepting… – user10101 Nov 01 '21 at 04:54
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    @ryang I believe “let” is fine as I see many people use this vernacular in their proofs. “for” is a condition statement, thus, is necessary. – user10101 Nov 01 '21 at 04:56
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    @user10101 (1) "let" is fine; I suggested to replace the second one only to distinguish the "forall" variable from the "exists" variable; it's not important and I did say it's just a small suggestion . (2) No, the construction "if/for [condition] implies that [consequent]" is ungrammatical; if you insist on the "if/for", then delete "implies that" (optionally, replace "implies that" with "then / one has / we have", like what this Answer does). – ryang Nov 01 '21 at 05:23
  • @user10101: ryang is wrong. The "Let ε>0." is common but very bad; see this post. – user21820 Aug 09 '22 at 12:13
  • "Let ε>0", in this context, is short for ""Let ε be an arbitrary positive number", and is absolutely standard and correct. The word "let" here is merely declaring a variable, while the unstated word "arbitrary" is performing existential instantiation; when wrapping up the proof, it is because ε had been arbitrary that we can then conclude, "for each ε...". – ryang Aug 27 '22 at 06:11