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Oct
27
comment Limit of $x_n/n$ as $n\to\infty$
@sundaycat: To take limit in $(5)$, I need assumption (ii) to apply $(1)$, where a reasonable requirement is $\theta_n$ cannot be too large. Without any assumption, I only know $0<\theta_n<\delta_n$. To control the upper bound of $\theta_n$, I need assumption $(1)$ to obtain $\delta_n\le\delta$ when $n$ is large. For example, consider $f(x)=x^p$($p\ge 1$) and an arbitrary $\theta:(0,\infty)\to (0,\infty)$. Then you need some condition on $\theta$ to get $\lim_{x\to\infty}\frac{f'(x+\theta(x))}{f'(x)}=1$.
Oct
26
comment Limit of $x_n/n$ as $n\to\infty$
@MartinArgerami: Thanks for your comment. I completely changed my argument after reading other answers.
Oct
26
revised Computing a limit almost surely using the strong law of large numbers
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Oct
26
awarded  Yearling
Oct
26
revised Limit of $x_n/n$ as $n\to\infty$
added 76 characters in body
Oct
25
comment Limit of $x_n/n$ as $n\to\infty$
From my answer you can see that letting $y_n=x_n^2$ is a good choice for your question (1) but not as good a choice for your question (2). The reason for letting $y_n=x_n^2$ in question (1) can be understood in this way: the associated function $f$ is question (1) is $c x^2$ for some $c>0$. For the same reason, in question (2), a good choice is letting $y_n=x_n^{3/2}$.
Oct
25
revised Limit of $x_n/n$ as $n\to\infty$
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Oct
23
revised Limit of $x_n/n$ as $n\to\infty$
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Oct
23
answered Limit of $x_n/n$ as $n\to\infty$
Oct
22
reviewed Leave Open Calculating index of a subgroup
Oct
22
comment Is the image of a nowhere dense closed subset of $[0,1]$ under a differentiable map still nowhere dense?
@NateEldredge: Here $E$ is closed and hence....
Oct
22
comment A Problem on Improper Integrals
@ParamanandSingh: Thank you for your tolerance.
Oct
22
comment A Problem on Improper Integrals
@ParamanandSingh: You are welcome! I still feel sorry about misleading you with a flawed argument. Maybe I I shouldn't post any answer when my head is not clear enough. Don't worry about the bounty; you can wait until the bounty expires to see if there is a better answer.
Oct
22
comment A Problem on Improper Integrals
@ziangchen: Thank you again for your appreciation.
Oct
22
comment A Problem on Improper Integrals
@ParamanandSingh: Terribly sorry! I mistakenly thought $h(0)=0$ before. I have updated my answer.
Oct
22
comment A Problem on Improper Integrals
@ziangchen: No, it was flawed, and now it's corrected. My brain is a little unclear today and I mistakenly thought $h(0)=0$ before. Thank you!
Oct
22
revised A Problem on Improper Integrals
added 54 characters in body
Oct
22
comment A Problem on Improper Integrals
@ParamanandSingh: I wrote my answer in this way because I thought you would prefer a hint rather than a full answer. Is it better if I add all the details to my answer?
Oct
22
comment For a closed plane curve, showing some inequalities.
@JeongNam-ho: You are welcome.
Oct
22
revised A Problem on Improper Integrals
added 150 characters in body