# Tagged Questions

Questions on the evaluation of limits.

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### What is the result of infinity minus infinity?

What is $\infty - \infty$? Is it $\infty$ or $0$ or what?
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### Proof of 1=0 by mathematical induction?

I got stuck with a problem that pop up in my mind while learning limits. I am still a high school student. $\lim\limits_{n\to\infty}(\underbrace{\frac{1}{n}+\frac{1}{n}+\cdots+\frac{1}{n}}_{m})=0$ ...
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### What it the fastest/most efficient algorithm for estimating Euler's Constant $\gamma$?

What it the fastest algorithm for estimating Euler's Constant $\gamma \approx0.57721$? Using the definition: $$\lim_{n\rightarrow\infty} \sum_{x=1}^{n}\frac{1}{x}-\log n=\gamma$$ I finally get 2 ...
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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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### $\lim_{n\rightarrow\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$

I'm supposed to calculate: $$\lim_{n\rightarrow\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}$$ By using W|A, i may guess that the limit is $\frac{1}{2}$ that is a pretty interesting and nice result. ...
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### Computing $\lim\limits_{n\to+\infty}n\int_{0}^{\pi/2}xf(x)\cos ^n xdx$

I got stuck at the following problem. Let $f\in C([0,\pi/2])$, then compute $$\lim_{n\to+\infty}n\int\limits_{0}^{\pi/2}xf(x)\cos ^n xdx$$ Could you suggest a helpful idea?
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### How to show that $\lim\limits_{x \to \infty} f'(x) = 0$ implies $\lim\limits_{x \to \infty} \frac{f(x)}{x} = 0$?

I was trying to work out a problem I found online. Here is the problem statement: Let $f(x)$ be continuously differentiable on $(0, \infty)$ and suppose $\lim\limits_{x \to \infty} f'(x) = 0$. ...
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### Calculating $\lim_{n\to\infty}\sqrt{n}\sin(\sin…(\sin(x)..)$

I was asked today by a friend to calculate a limit and I am having trouble with the question. Denote $\sin_{1}:=\sin$ and for $n>1$ define $\sin_{n}=\sin(\sin_{n-1})$. Calculate ...
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### limit of : $a_{n+2} =\frac{1}{a_n} + \frac{1}{a_{n+1}}$

$a_n$ is a real sequence, $a_1,a_2$ are positive and for all $n>2$ : $$a_{n+2} =\frac{1}{a_{n+1}} + \frac{1}{a_{n}}.$$ Prove that: $\displaystyle \lim_{n\to \infty} a_n$ exists, then find it. ...
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### An interesting sum to infinity

Is there any simple way of computing the following sum? $$\sum_{k=1}^\infty \frac1{k\space k!}$$
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### The limit of $\frac{n+\sqrt{n}+\cdots+\sqrt[n]{n}}{n}$

How do I compute the following limit or show it doesn't exists? $$\lim_{n\rightarrow\infty}\frac{n+\sqrt{n}+\cdots+\sqrt[n]{n}}{n}$$ I've struggled with this problem for a while now so I would ...
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### Is there a step by step checklist to check if a multivariable limit exists and find its value?

Do we rely on certain intuition or is there an unofficial general crude checklist I should follow? I had a friend telling me that if the sum of the powers on the numerator is smaller then the ...
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### Three sequences and a limit(own)

Let us consider three sequences $(a_n)_{n\ge1}$, $(b_n)_{n\ge1}$ and $(c_n)_{n\ge1}$ having the properties: $a_{n},\ b_{n},\ c_{n}\in\left(0,\ \infty\right)$ ...
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### Limit of sum with binomial coefficient

Prove that: $$\lim_{n\to +\infty}\sum_{k=0}^{n}(-1)^k\sqrt{{n\choose k}}=0$$ I completely don't know how to approach. Is it very difficult?
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### $\lim\limits_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite

How do I prove that $\displaystyle\lim_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite?
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### Compute: $\lim_{n\to\infty}\frac{{\left(1^1 \cdot 2^2 \cdot3^3\cdots n^n\right)}^\frac{1}{n^2}}{\sqrt{n}}$

Compute the following limit: $$\lim_{n\to\infty}\frac{{\left(1^1 \cdot 2^2 \cdot3^3\cdots n^n\right)}^\frac{1}{n^2}}{\sqrt{n}}$$ I'm interested in almost any approaching way for this limit. ...
### 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 ...