# Tagged Questions

Questions on the evaluation of limits.

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### Epsilon Limit Proof (Bridge to Abstract Mathematics)

For $(3n)$, how would I prove that the limit of $n$ approaching infinity does not exist? Obviously this would diverge but I'm not completely sure how to prove it. I know I have to use epsilon delta. ...
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### What does $\lim_{\epsilon\to 0} \frac{\zeta(1+\epsilon) + \zeta(1-\epsilon)}{2} =\gamma$ really mean?

$$\lim_{\epsilon\to 0} \frac{\zeta(1+\epsilon) + \zeta(1-\epsilon)}{2} =\gamma$$ I am somewhat familiar with the zeta function, but have not taken complex analysis, yet. I saw this on Wikipedia and ...
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### Does the following sequence converge or diverge? [closed]

Does the following sequence converge or diverge? $$\lim_{n \to \infty} \sqrt{16n+2} - \sqrt{16n}$$
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### How can I calculate $\displaystyle\lim_{x \to \infty}x^{2}\ln\left(\cos \left(\pi/x\right)\right)$?

Does anybody know how to solve this? $$\lim_{x \to \infty}x^{2}\ln\left(\cos\left(\pi \over x\right)\right)$$
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### Limit $\lim_{n\to\infty}n\frac{\sin\frac{1}{n}-\frac{1}{n}}{1+\frac{1}{n}}$

I need to find a limit of a sequence: $$\lim_{n\to\infty}n\frac{\sin\frac{1}{n}-\frac{1}{n}}{1+\frac{1}{n}}$$ I tried to divide numerator and denominator by n, but it didn't help, as the limit became ...
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### Find a limit $\lim_{x \to - \infty} \left(\frac{4^{x+2}- 2\cdot3^{-x}}{4^{-x}+2\cdot3^{x+1}}\right)$ [duplicate]

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 ...
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### How to prove geometrically the limit $\lim_{x \to 0}\frac{1-\cos{x}}{x}$ using squeeze theorem [duplicate]

Here, I ask how to geometrically prove the limit $$\lim_{x \to 0}\frac{1-\cos{x}}{x}$$ using squeeze theorem. I see so many proofs for the other important limit ...
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### Expectation involving a maximum of a sequence of i.i.d. Gaussians

Let $X_1,\ldots,X_n$ be a sequence of i.i.d. standard Gaussian random variables. Denote the maximum of this sequence by $M_n$. I am interested in evaluating the following expectation: ...
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### Question about $f$ continuous function with these conditions?

Suppose I have a differentiable and bounded function $$f: [0, + \infty) \longrightarrow \mathbb{R}$$ such that $$\forall x \in [0, + \infty) \, : f(x) \cdot f'(x) > \sin x.$$ The question is: ...
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### Can someone guide me through these two convergence & divergence problems?

$A_n = \cos(\frac{n\pi}{2})$ $A_n = \frac{\ln(3n)}{\ln(n)}$ So, for the first one, it looks like I'll have to use the squeeze theorem. $-1\le \cos(\frac{n\pi}{2}) \le 1$ I'm slightly ...
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### Real Analysis, limit of function using binomial theorem.

We have $b \in \mathbb{R}$ and $0<b<1$, I need to show that $\lim(nb^n)=0$ My attempt: Since $0<b<1$ we can express b as $\frac{1}{1+a_n}$ so now our sequence is ...
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### For a continuous function $f$ and a convergent sequence $x_n$, lim$_{n\rightarrow \infty}\,f(x_n)=f(\text{lim}_{n \rightarrow \infty} \, x_n)$

Let $f:X \rightarrow Y$ be a function. Prove that if $f$ is continuous, then for every convergent sequence $(x_n)$ lim$_{n\rightarrow \infty}\,f(x_n)=f(\text{lim}_{n \rightarrow \infty} \, x_n)$ My ...
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### Find $\lim_{x \to - \infty} \left(\frac{4^{x+2}- 2\cdot3^{-x}}{4^{-x}+2\cdot3^{x+1}}\right)$

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 ...
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### Complex differentiability equivalent to linear approximation

Let $G \subset \mathbb C$ be an open set and $f: G \to \mathbb C$ a complex function on $G$. Prove that the function $f$ is complex differentiable at a point $z \in G$ if and only if there exists a ...
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### Differentiation, from first principles

I am having problems with this question, it would be wonderful if someone can help. Given that $f(x)= x^2 + x - 3$ 1) Find $f(x + h)$ 2) Then express $f(x+h)-f(x)$ in its simplest form 3) Deduce ...
### Does convergence of $(a_{5n})$ and $(a_{n+1} - a_n)\rightarrow0$ imply convergence of $a_n$
As I wrote in the title: Does convergence of $(a_{5n})$ and the fact that $(a_{n+1} - a_n)\rightarrow0$ imply convergence of $a_n$. I understand that convergence of $(a_{5n})$ means that $(a_n)$can ...