# Tagged Questions

Questions on the evaluation of limits.

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### How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?

How can one prove the statement $$\lim\limits_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution. This is homework. In my ...
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### $f$ uniformly continuous and $\int_a^\infty f(x)\,dx$ converges imply $\lim_{x \to \infty} f(x) = 0$

Trying to solve $f(x)$ is uniformly continuous in the range of $[0, +\infty)$ and $\int_a^\infty f(x)dx$ converges. I need to prove that: $$\lim \limits_{x \to \infty} f(x) = 0$$ Would ...
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### How to prove that exponential grows faster than polynomial?

In other word, how to prove: For all real constants $a$ and $b$ such that $a > 1$, $$\lim_{n\rightarrow\infty}\frac{n^b}{a^n} = 0$$ I know the definition of limit but I feel that it's ...
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### What is the limit of $n \sin (2 \pi \cdot e \cdot n!)$ as $n$ goes to infinity?

I tried and got this $$e=\sum_{k=0}^\infty\frac{1}{k!}=\lim_{n\to\infty}\sum_{k=0}^n\frac{1}{k!}$$ $$n!\sum_{k=0}^n\frac{1}{k!}=\frac{n!}{0!}+\frac{n!}{1!}+\cdots+\frac{n!}{n!}=m$$ where $m$ is an ...
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### Uses of $\lim \limits_{h\to 0} \frac{f(x+h)-f(x-h)}{2h}$

I have been wondering whether the following limit is being used somehow, as a variation of the derivative: $$\lim_{h\to 0} \frac{f(x+h)-f(x-h)}{2h} .$$ Edit: I know that this limit is defined in ...
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### On the Diophantine equation $a^2+b^2 = c^2+k$

Given the Diophantine equation, $$a^2+b^2=c^2+k$$ where k is a constant integer. Let $0 < a \le b$, and $\Delta_k(N)$ be the number of primitive solutions with $0 < c < N$ for some bound ...
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### Basic facts about ultrafilters and convergence of a sequence along an ultrafilter

Could you help, please. I need the information about the ultrafilters, namely, any ideas how one can see that they exist and a proof of the fact that for any ultrafilter every sequence on a compact ...
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### How to prove that $\lim \limits_{n\rightarrow \infty} \frac{F_{n+1}}{F_n}=\frac{\sqrt{5}+1}{2}$

How would one prove that $$\lim_{n\rightarrow \infty} \frac{F_{n+1}}{F_n}=\frac{\sqrt{5}+1}{2}=\varphi$$ where $F_n$ is the nth Fibonacci number and $\varphi$ is the Golden Ratio?
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### Why does this limit exist $x^{x}$

I was wondering what is the value of $\lim\limits_{x\to 0+} x^{x}$. Assuming that the limit exists, I could show using the usual logarithm techniques that the limit is $1$. However, I am not able to ...
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### Proving that $\lim\limits_{x \to 0}\frac{e^x-1}{x} = 1$

I was messing around with the definition of the derivative, trying to work out the formulas for the common functions using limits. I hit a roadblock, however, while trying to find the derivative of ...
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### Finding $\lim\limits_{h \to 0}\frac{(\sqrt{9+h} -3)}{h}$

I have about 10 of these problems I could post, can't figure any of them out...not sure what to do. $$\lim_{h \to 0}\dfrac{(\sqrt{9+h} -3)}{h}$$ I know that I need to rationalize it so I multiply by ...
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### Does $\sum\limits_{k=1}^n 1 / k ^ 2$ converge when $n\rightarrow\infty$?

I can prove this sum has a constant upper bound like this: $$\sum_{k=1}^n \frac1{k ^ 2} \lt 1 + \sum_{k=2}^n \frac 1 {k (k - 1)} = 2 - \frac 1 n \lt 2$$ And computer calculation shows that sum ...
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### Does the inverse of this matrix of size $n \times n$ approach the zero-matrix in the limit as $\small n \to \infty$?

Fiddling with another (older) question here I constructed an example-matrix of the type $\small M_n: m_{n:r,c} = {1 \over (1+r)^c } \quad \text{ for } r,c=0 \ldots n-1$ . I considered the inverse ...
I was musing about a particular limit, $L = \prod\limits_{n > 0} \bigl(1 - 2^{-n}\bigr)$: we may bound 0.288 < L < 0.308, which we may show by taking the logarithm: ...
Evaluate: $$\lim_{x\to0^+}\frac{(-3\sqrt{x}+x^3+\sin(x^6))(\text{e}^{4\sqrt{x}}-1)}{4\ln (1+x)+x^4-x^6}$$