# Tagged Questions

Questions on the evaluation of limits.

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### calculate the $lim_{n\to\infty} V_{n}$

How would I calculate $lim_{n\to\infty} V_{n}(\forall n\in \mathbb{N}), V_{n}=(\sum_{k=0}^{2n}\sqrt{k})-(2n+1)\sqrt{n}$
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### Find $\lim_{x\to\infty} \frac{(x+1)^2(2x-4)^3}{(2x-1)^4}$

Find $$\lim_{x\to\infty} \frac{(x+1)^2(2x-4)^3}{(2x-1)^4}$$ I want to divide by $x^4$, but wouldn't I need to multiply out the whole thing and make a mess? Is there a more simple way of doing this? ...
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### Finding the limit of $\left(\frac{n}{n+1}\right)^n$

Find the limit of: $$\lim_{n\to\infty}\left(\frac{n}{n+1}\right)^n$$ I'm pretty sure it goes to zero since $(n+1)^n > n^n$ but when I input large numbers it goes to $0.36$. Also, when ...
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### limit of $x \log y$ at $(0,0)$

What is the limit of the function $x \log y$ at $(0,0)$? I believe the limit doesn't exist. But wolframalpha.com says the limit is 0. ...
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### Why does $\lim_{x \rightarrow 0} B(x,y)$ exist and how is it calculated?

In evaluating integrals like (link to another example) $$I=\int_0^1\frac{\log(x) \log^2(1-x)dx}{x}$$ one can make the substitution $x=\sin^2(\theta)$ to obtain ...
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### L'Hopital's rule question [on hold]

How do you evaluate the limit $$\lim_{x \to 0^+} \left(x^{\tan x} + x^2 \frac{\sin(1/x)}{\tan x}\right)?$$ The answer is 1. (the first part is 1 and the second part is 0.) Thanks in advance!
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### how do you read: “$\lim (n-1)/(n-2) = 1$”

The limit of the fraction $(n-1)/(n-2)$ with $n$ approaching $+\infty$ is $1$. But how do you read that, concisely? (say during chalking the formula on a blackboard). Is it acceptable to say "lim", ...
I'm stuck on some scattering homework, could anyone help me with this limit involving Legendre polynomials? $$\lim_{\theta\to 0}\frac{P_n^1 (\cos \theta)}{\sin \theta}$$