Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [limits]

Questions on the evaluation and properties of limits in the sense of analysis and related fields. For limits in the sense of category theory, use the tag “limits-colimits” instead.

In mathematics, a limit is the value that a function or sequence "approaches" as the input or index approaches some value. Limits are essential to (and mathematical analysis in general) and are used to define continuity, derivatives, and integrals.

The formal $\varepsilon$-$\delta$ definition of a finite limit at a point $a\in \mathbb{R}$ is:

$$\Big(\lim_{x\rightarrow a} f(x) = L \Big)\iff \Big(\forall \varepsilon >0\, \exists \delta > 0: \forall x\in D\quad 0<\vert x-a\vert <\delta \implies \vert f(x)-L\vert <\varepsilon \Big).$$

The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to the concepts of limit and direct limit in category theory.

43700 questions
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Compute $\lim_{n \to \infty}(1+\frac{n-1}{n+1})^{\frac{n+1}{n-1}}$

Compute $\lim_{n \to \infty}(1+\frac{n-1}{n+1})^{\frac{n+1}{n-1}}$ I did: $\lim_{n\to \infty}(1+\frac{1}{\frac{n+1}{n-1}})^{\frac{n+1}{n-1}}=e$. Why is this incorrect? Thank you for your help.
J.Doe
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Question about limits

I would like to know what would be the best procedure to evaluate the limits of the following functions; some explanation would be appreciated: $$\lim_{\theta\rightarrow -\infty}\dfrac{\cos\theta}{3\theta}$$ and $$\lim_{\theta\rightarrow…
azetina
  • 229
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Find the limit of $e^x/2^x$ as $x$ approaches infinity

I am trying to find the asymptotic relation between $e^x$ and $2^x$. I tried to use limit comparison: $$\lim_{x\to\infty}\left(\frac{e^x}{2^x}\right)$$ I tried to use L'Hopital's…
He Yifei 何一非
  • 153
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Evaluate $\lim\limits_{x\to \infty} \frac {x^x-4}{2^x-x^2}$

Evaluate $$\lim_{x\to \infty} \frac {x^x-4}{2^x-x^2}$$ I think it needs to use L'Hospital Rule. So, I first calculate $\frac {d x^x}{dx}= x^x(\ln x+1)$. And then $$\lim_{x\to \infty} \frac {x^x-4}{2^x-x^2}=\lim_{x\to \infty} \frac {x^x(\ln…
Maggie
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Find the limit of $\left(1+ \frac{2}{n}\right)^{n^{2}} \exp(-2n)$ as $n \to \infty$.

Find the limit of $\left(1+ \frac{2}{n}\right)^{n^{2}} \exp(-2n)$ as $n \to \infty$. By expansion - $$\lim\limits_{n \to \infty} \left[1+(n^{2})(2/n) + (n^{2})(n^{2}-1)/2 \dots ]/[1+2n+(2n)^{3}/3! \dots\right]$$ I didn't get any result. By…
Mathaddict
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Limit Question $\lim_{x\to\infty} \sqrt{x^2+1}-x+1$

I understand the answer is 1 which kind of makes sense intuitively but I can't seem to get there. I would appreciate if someone pointed out which line of my reasoning is wrong, thanks. I tried writing all my steps \begin{equation} …
Quaz
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Using the Squeeze Theorem on $\lim_{x\to 0}\frac{\sin^2x}{x^2}$

$$\lim_{x\to 0}\frac{\sin^2x}{x^2}$$ I'm trying to evaluate this limit using Squeeze Theorem. However, looking at the graph I know it approaches $1$, but I am getting $0$ using the Squeeze Theorem. $$-\frac{1}{x^2} < \frac{\sin^2x}{x^2} <…
user592234
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Does $\lim\limits_{x\to\infty}$ equal to $\lim\limits_{x\to+\infty}$?

As per title, does $$\lim\limits_{x\to\infty}$$ mean $\lim\limits_{x\to+\infty}$ or $\lim\limits_{x\to\pm\infty}$? This link seems to tell me that it's the latter: https://qc.edu.hk/math/Certificate%20Level/Limit%20mistake.htm However, evaluating…
Loo Soo Yong
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How to prove that $\lim_{n\rightarrow\infty}\left[\frac{(n+1)^{n+1}}{n^n}-\frac{n^n}{(n-1)^{n-1}}\right] = e$

I learnt on Wolfram MathWorld that $$\lim_{n\rightarrow\infty}\left[\frac{(n+1)^{n+1}}{n^n}-\frac{n^n}{(n-1)^{n-1}}\right] = e$$ How should I prove…
Larry
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Limit Question: doubting about my answer! A limit where $x \to \infty$

Hey guys so I have this limit: $$\lim_{x \to ∞} f(x) = \frac{(3x+1)^3(x-1)}{(x-2)^4}$$ and I got $27$ as final answer, just wondering if you guys can check; I expanded the numerator and denominator and then divided everything by $x^4$
S..
  • 109
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Does $\lim_{n\to\infty}\frac{3^n+5^n}{(-2)^n+7^n}$ exist?

My textbook says that this limit doesn't exist, but I don't understand - why? I tried calculating it by taking from both numerator and denominator factors that diverge to $\infty$ the fastest: $$\lim_{n\to\infty}\frac{3^n+5^n}{(-2)^n+7^n}=…
agromek
  • 51
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limit of $\sqrt{x^6}$ as $x$ approaches $-\infty$

I need to solve this limit: $$\lim_{x \to - \infty}{\frac {\sqrt{9x^6-5x}}{x^3-2x^2+1}}$$ The answer is $-3$, but I got 3 instead. This is my process: $$\lim_{x \to - \infty}{\frac {\sqrt{9x^6-5x}}{x^3-2x^2+1}} = \lim_{x \to - \infty}{\frac…
Netanel
  • 377
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$\lim_{n\to\infty} n^{x}(a_{1}a_{2}\ldots a_{n})^{\frac{1}{n}}=ae^x$

Let $ (a_{n})$ be positive sequence, $a,x \in R \quad $ and $ \lim_{n\to\infty} n^{x}a_{n}=a$. Prove that $\lim_{n\to\infty} n^{x}(a_{1}a_{2}\ldots a_{n})^{\frac{1}{n}}=ae^x$ I know that $\lim_{n\to\infty} (a_{1}a_{2}\ldots…
math.trouble
  • 123
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Limit of a sequence with n root

I have a trouble with this example: $$n(\sqrt[n]n-1)$$ I've been trying to do it this this way: $$a_n = \frac{(\sqrt[n]n-1)(\sqrt[n]n^{n-1}+\sqrt[n]n^{n-2}+\dots+1)}{\sqrt[n]n^{n-1}+\sqrt[n]n^{n-2}+\dots+1}=…
user609637
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