Evaluate $\displaystyle\lim_{x\to\infty}\frac{(\ln x)^n}x$, where $n\in\Bbb N$.   I think $(\ln x)^n$ &amp; $x$ is positive infinity when $x$ goes positive infinity so use a L'Hospital's theorem
124062
Find $\displaystyle \lim_{x \to \infty} \dfrac{x}{(\log{x})^n}$.   My book says that $\displaystyle \lim_{x \to \infty} \dfrac{(\log{x})^n}{x} = \lim_{y \to \infty} \dfrac{y^n}{e^y} = 0$, but I don't see how that is true.
21873
Computing $\displaystyle \lim_{n\rightarrow \infty}\left(\frac{1^n+2^n+\cdots \cdots +n^n}{n^n}\right)$  Attempt: $\displaystyle \lim_{n\rightarrow \infty}\bigg[\left(\frac{1}{n}\right)^n+\left(\frac{2}{n}\right)^n+\cdots \cdots +\left(\frac{n-1}{n}\right)^n+\left(\frac{n}{n}\right)^n\bigg] = 1$  Is my answer right? If not, then could someone help me? Thanks.
28260
Assume that $$\lim_{n\to\infty}(a_{n+2}-a_{n})=A$$ show that  $\displaystyle \lim_{n\to\infty}\dfrac{a_{n}}{n}$and $\displaystyle\lim_{n\to\infty}\dfrac{a_{n+1}-a_{n}}{n}$ exist and find these limits.  maybe this problem have some methods, Thank you.  since  $$\lim_{n\to\infty}(a_{n+2}-a_{n})=A$$ so there exists $N$, and for $\forall \varepsilon&gt;0$, such $$|a_{n+2}-a_{n}-A|\le\varepsilon$$
27188
I'm being asked to determine if $\displaystyle\sum\limits_{n=3}^\infty \frac1{n (\ln n)\ln(\ln n)}$ converges. So, using Cauchy's Condensation Test, I reduced the problem to one of determining the convergence of $\displaystyle\sum\limits_{n=3}^\infty\frac 1{n\ln (n\ln 2)}$. Am I on the right path, and how do I proceed from here?
57939
$$\lim_{n \to \infty }n\int_{0}^{1}\frac{x^{n}}{1+x+x^{n}}dx$$  I factorize $x^n$ then I tried with $u=1+x^{1-n}$ but I didn't get too far. Also I tried to get something from $0&lt;x&lt;1$. I know that if $x$ is from $(0,1)$ then $x^n$ tends to $0$ as $n \to \infty$.
31748
How to find the value of $\lim_{n\to\infty}S(n)$, where $S(n)$ is given by $$S(n)=\displaystyle\sum_{k=1}^{n} \dfrac{k}{n^2+k^2}$$   is unable to calculate it.  This is a question from a questions booklet, and the options for the answer are--  $\begin{align} &amp;A) \dfrac{\pi}{2} \\ &amp;B) \log 2 \\ &amp;C) \dfrac{\pi}{4} \\ &amp;D) \dfrac{1}{2} \log 2 \end{align}$
75170
Let $\{x_n\}_{n=1}^\infty$ be a squence of nonnegative real numbers such that $\displaystyle \lim_{n \to \infty} x_n =x \in \Bbb R$. If $\alpha \in (0, \infty)$, is it true that $\displaystyle \lim_{n \to \infty} x_n^\alpha =x^\alpha$?  I read the proofs for $\alpha$ integer and for $\alpha=1/2$, but I wonder what happens with the other cases.
91101
Use Taylor's theorem to prove that $\displaystyle\lim_{n \to \infty} n \ln\left(1+\frac{1}{n}\right)=1$  I don't understand how to apply Taylor's theorem to a limit, especially one with a product of two functions...
100451
I have a very simple question. Suppose I want to evaluate this limit:  $$\lim_{x\to \infty} \frac{x}{x-\sin x}$$  It is easy to evaluate this limit using the Squeeze theorem (the answer is $1$). But here both the numerator and the denominator are going to infinity as $x\to \infty$ so I tried using L'Hospital's rule: $$\lim_{x\to \infty} \frac{x}{x-\sin x}=\lim_{x\to \infty} \frac{1}{1-\cos x}$$  However there's no finite $L$ such that $$\lim_{x\to \infty} \frac{1}{1-\cos x}=L$$ which is a contradiction. I don't understand why in this case L'Hopital's rule doesn't work. Both the numerator and the denominator are differentiable everywhere and both are tending to infinity - which is all we need to use this rule.
92775
Let $\alpha$ be a positive number. Calculate $$\lim_{n\to\infty} \sum_{k=1}^{n} \Big(\frac{k}{n}\Big)^{\alpha k}$$  Edit: I have deleted my attempt, it didn't seem to lead me anywhere and I discovered a couple of mistakes. :)  Please give me a hint!
109450
$$\lim_{x\to y}\frac{x^y-y^x}{x^x-y^y} $$ I've tried L'Hospital and  Both doesn't tends to work. Please help someone.
82086
Find the following limit:  $$\lim_{n\to \infty} (1+n)^{\frac1n}$$  I know that  $\lim_{n\to 0} (1+n)^{\frac1n}= e$.
80661
Given function $g$ continuous in line $[0,\infty)$, positive and strongly increasing that upholds $\lim_{t\to\infty}  g(t) = L$. Calculate the limit:  $$ \lim_{n\to\infty} \int\limits_1^2 g(nx) \, dx. $$  At the moment what I think that I should do is swap the $g(nx)$ with $L$ as it approaches infinity but from there on I am just not sure whats my next step is.
58830
I have to find the radius of convergence for $\displaystyle\sum_{n=0}^{\infty}\frac{n}{n+1}x^n$, for x$\in$R. I think it is $r^{-1}=\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=\lim_{n\to\infty}\frac{ \frac{n+1}{n+2} }{ \frac{n}{n+1}}=1$ so r=1. Then I have to find out if $\displaystyle\sum_{n=0}^{\infty}\frac{n}{n+1}x^n$ is convergent in  the end points of the convergence interval. I think I have to check for $|r|$. But we can't use ratio test (L=1) or integral test(not decreasing) for $\displaystyle\sum_{n=0}^{\infty}\frac{n}{n+1}$. How can I then check if it's convergent in the end points of the convergence interval?
52861
Finding $\displaystyle \lim_{y\rightarrow \infty}\bigg[\frac{y}{e}-y\bigg(\frac{y}{y+1}\bigg)^y\bigg]$   Attempt: Put $\displaystyle y=\frac{1}{z}$ . Then $\displaystyle \lim_{z\rightarrow 0}\bigg[\frac{1}{ze}-\frac{1}{z}\bigg(\frac{z}{z+1}\bigg)^{\frac{1}{z}}\bigg]$  Could some help me to solve it, thanks
84537
find value of the $$\lim_{x\to0}\frac{e-(1+x)^{\frac{1}{x}}}{x}$$I use hospital law and can't find answer
95810
How do I evaluate this limit: $$\lim\limits_{x\to \infty}\frac{\sin x}{x}$$  Is it $0$? If yes, how so?