Problem : Compute the following limit: $ \lim_{n\rightarrow \infty } {\frac{1^{1}+2^{2}+3^{3}+...+n^{n}}{n^{n}}}$  I have known the solutions by using  Stolz's Law. Then I try to compute it by following Squeeze Theorem, but it seems hard for me to find upper bound.  Please help me! Thank you very much!
35981
$$\lim_{n \to \infty}{\frac{1^1+2^2+3^3+\cdots+n^n}{n^n}}.$$  With a first look this must give $1$ as a result but have a problem to explain it.  How can I do it?  Edit  I noticed that it is $\frac{\infty}{\infty}$.  $$\lim_{n \to \infty}{n^{n}\frac{(\frac{1^1}{n^{n}}+\frac{2^2}{n^{n}}+\frac{3^3}{n^{n}}+\cdots+1)}{n^n}}= \lim_{n \to \infty}{\frac{1^1}{n^{n}}+\frac{2^2}{n^{n}}+\frac{3^3}{n^{n}}+\cdots+1}=1$$  Is this correct?
101463
Find the limit $$\lim_{n \to \infty}\left[\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\cdots\left(1-\frac{1}{n^2}\right)\right]$$   I take log and get $$\lim_{n \to \infty}\sum_{k=2}^{n} \log\left(1-\frac{1}{k^2}\right)$$
60294
How to solve the following limit question? $$\lim_{n\to\infty}\frac{1}{e^n}\Bigl(1+\frac1n\Bigr)^{n^2}$$  Thanks a lot.
112932
Any hints on how to find the following limit ?.  I haven't been able to figure it out still. $$ \lim_{n \to \infty}\left(\, 1 - \frac{1}{3}\, \right)^{2} \left(\, 1 - \frac{1}{6}\, \right)^{2}\left(\, 1 - \frac{1}{10}\, \right)^{2}\ldots\left[\, 1 - \frac{1}{n\left(\, n + 1\, \right)/2}\, \right]^{2} $$  Pardon me if it's a silly question.  Edit: Solved upto this much   $$ \prod_{n = 2}^{\infty}\left[\, 1-\frac{2}{n\left(\, n + 1\, \right)}\, \right] = \prod_{n = 2}^{\infty}\left(\, \frac{n + 2}{n + 1}\,\frac{n - 1}{n}\, \right) $$  I did'nt notice most of the terms cancel out.
41326
I am to find the limit of  $$\lim_{x \to - \infty} \left(\frac{4^{x+2}- 2\cdot3^{-x}}{4^{-x}+2\cdot3^{x+1}}\right)$$  so I used: $$\lim_{x \to -\infty} = \lim_{x \to \infty}f(-x)$$  but I just can't solve it to the end.
19643
The question is rather:      Prove the following:   $$  \lim_{x \to \infty} \frac{x ^ {\sqrt x} }{ 2 ^ x} = 0. $$   I was thinking of using the Squeeze Theorem (might not be the right way to go), but finding an upper-bound function proved to be quite tricky.
9283
How can I compute the limit $\lim_{n\rightarrow\infty}\left(1+\frac{1}{n}\right)^{n^2}$? Of course $\lim_{n\rightarrow\infty}\left(1+\frac{1}{n}\right)^{n} = e$, and then $\left(1+\frac{1}{n}\right)^{n^2} = \left(\left(1+\frac{1}{n}\right)^n\right)^n$. Since the term inside converges to $e$, the whole thing is $e^n$, and can I conclude that the limit is infinity as $n\rightarrow\infty$?
48528
I want to prove the following limit:  $$\lim_{n \to \infty} \sqrt[n+1\;] {(n+1)!} - \sqrt[n] {(n)!} = \frac{1}{e}.$$  I searched the forum &amp; found the link here:  .  But still, there is no way out of the problem. So, please solve it.
113515
Problem Evaluate  $$\lim\limits_{n\rightarrow \infty}\frac{n+n^2+n^3+\cdots +n^n}{1^n+2^n+3^n+\cdots +n^n}.$$ My solution Notice that $$\lim_{n \to \infty}\frac{n+n^2+n^3+\cdots +n^n}{n^n}=\lim_{n \to \infty}\frac{n(n^n-1)}{(n-1)n^n}=\lim_{n \to \infty}\frac{1-\dfrac{1}{n^n}}{1-\dfrac{1}{n}}=1,$$and  $$\lim_{n \to \infty}\frac{1+2^n+3^n+\cdots+n^n}{n^n}=\frac{e}{e-1}.$$  Hence,\begin{align*}\lim\limits_{n\rightarrow \infty}\frac{n+n^2+n^3+\cdots +n^n}{1^n+2^n+3^n+\cdots +n^n}&amp;=\lim_{n \to \infty}\frac{\dfrac{n+n^2+n^3+\cdots +n^n}{n^n}}{\dfrac{1+2^n+3^n+\cdots +n^n}{n^n}}\\&amp;=\frac{\lim\limits_{n \to \infty}\dfrac{n+n^2+n^3+\cdots +n^n}{n^n}}{\lim\limits_{n \to \infty}\dfrac{1+2^n+3^n+\cdots +n^n}{n^n}}\\&amp;=1-\frac{1}{e}.\end{align*} The solution posted above need to quote an uncommon limit. Is there another more simple and more direct solution?
34560
$ \lim_{n\to\infty} {\sqrt{1+{\sqrt{2+{\sqrt{\cdots +\sqrt{n}\ }\ }\ }\ }\ \ }\ } = ? $ Either closed answer or an upper bound would help.
121778
$$\lim_{x\rightarrow 1} \frac{x^m-1}{x^n-1}$$   I found this in a book (recommended by the professor) without solutions or hints.  My problem is that I can't find definitions about this kind of problems and I have no idea how to start, any kind of help would be great. So far I've been dealing with "$\lim_{x\rightarrow ∞}$" only. Also without L'Hospital, because we haven't had this in lecture.
64177
I have this limit: $$\lim_{n\rightarrow\infty}\sin\left(\pi\sqrt[3]{n^{3}+1}\right)$$ I don't even know if it exists. If so, what its value ? Really don't have any idea..
9394
Find the following limit  $$ \lim_{x\to0}\left(\frac{1+x2^x}{1+x3^x}\right)^\frac1{x^2} $$  I have used natural logarithm to get  $$ \exp\lim_{x\to0}\frac1{x^2}\ln\left(\frac{1+x2^x}{1+x3^x}\right) $$  After this, I have tried l'opital's rule but I was unable to get it to a simplified form.  How should I proceed from here? Any here is much appreciated!
64635
Find $$\lim_{n \to \infty} \left( \frac{3^{3n}(n!)^3}{(3n)!}\right)^{1/n}$$ I don't know what method to use, if  we divide numerator and denominator with $3^{3n}$, I don't see that we win something. I can't find two sequences, to use than the squeeze theorem. I'm stuck.
62524
strong textFind $\lim_\limits{n\to \infty}\left({1\over \sqrt{n^2+1}}+{1\over \sqrt{n^2+2}}+\cdots+{1\over \sqrt{n^2+n}}\right)$. I do know it is bounded by $1$. I tried using the sandwich rule with no success. How can I solve it?
32524
How can I find the $\lim  (1+(a^{1/n}-1) / b)^n$ when $n \to \infty$?    I would greatly appreciate it if you kindly give me some hints.
28565
Find the limit $\displaystyle\lim_{n\rightarrow\infty}{(1+1/n)^{n^2}e^{-n}}$?  I found the limit as $e^{-1/2}$ using l'Hospital rule. I guess I made a mistake. Because the limit seems to be 1. Also, can we find the limit without L'Hospital rule?