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.

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Evaluating $\lim_{\theta \to 0^+}\frac{\sin\theta}{\theta^2}$

How do I evaluate $$\lim_{\theta \to 0^+}\frac{\sin\theta}{\theta^2}?$$ I tried the following: $$\lim_{\theta \to 0^+}\frac{\sin\theta}{\theta^2} = \lim_{\theta \to 0^+}\frac{1}{\theta}\cdot \lim_{\theta \to 0^+}\frac{\sin\theta}{\theta} =…
okarin
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Evaluating $\lim\limits_{x\mathop\to\infty}\frac{\tan x}{x}$

I need to find $$\lim\limits_{x\mathop\to\infty}\frac{\tan x}{x}$$ For some reason mathematica just returns my input without evaluating it. For what it's worth, $\dfrac{\tan(10^{100})}{10^{100}}\approx -4\times10^{-101}$, so the limit is probably…
user85798
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Find limit $\lim_{x\to\pi/2}\frac{\sin x - (\sin x)^{\sin x}}{1 - \sin x + \ln(\sin x)}$

Find the limit of $$\lim_{x\to\pi/2}\frac{\sin x - (\sin x)^{\sin x}}{1 - \sin x + \ln(\sin x)}.$$ Please help, D L rule does not help
user2369284
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Limit of fractional part

Prove that the limit as n tends to infinity from $\{n!\sqrt2\}$ does not exist. where {} denotes fractional part and "!" denotes factorial. I don't have many ideas. I would try to show that , given an arbitrary term $x_n$ we can find another term…
Ion
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How find this strange limit

find this follow strange limit $$\lim_{x\to 0}\dfrac{\arcsin{\arctan{\sin{\tan{\arcsin{\arctan{x}}}}}}-\sin{\tan{\arcsin{\arctan{\sin{\tan{x}}}}}}}{\arctan{\arcsin{\tan{\sin{\arctan{\arcsin{x}}}}}}-\tan{\sin{\arctan{\arcsin{\tan{\sin{x}}}}}}}\cdots…
math110
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How find this$\lim_{n\to\infty}n^2\left(n\sin{(2e\pi\cdot n!)}-2\pi\right)=\frac{2\pi(2\pi^2-3)}{3}$

show that $$\lim_{n\to\infty}n^2\left(n\sin{(2e\pi\cdot n!)}-2\pi\right)=\dfrac{2\pi(2\pi^2-3)}{3}$$ we are kown that $$\lim_{n\to\infty}n\sin{(2\pi e\cdot n!)}=2\pi$$ because we…
math110
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What is the value of $\lim \limits_{n\to \infty} \left(\sin\left(n\pi\sqrt[3]{n^3+3n^2+4n-5}\right)\right)$?

This question was posted recently on a Facebook group. Find the limit: $$\lim \limits_{n\to \infty} \left(\sin\left(n\pi\sqrt[3]{n^3+3n^2+4n-5}\right)\right)$$ My original analysis was: Regarding the term inside the cube root, as $n \to \infty$,…
Adrian
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From $\sqrt{2\pi}$ to Glaisher-Kinkelin to what?

I was reading about Glaisher-Kinkelin Constant and came across the following formulas $$ \sqrt{2\pi} = \lim_{n\rightarrow\infty} \frac{n!}{n^{\frac{2n+1}{2}}e^{-n}} $$ $$ A = \lim_{n\rightarrow\infty}…
gist076923
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Why is it wrong to solve the limit this way?

Consider the following limit： \begin{align*} &\lim_{x\to +\infty}\frac{e^x}{\left(1+\frac1x\right)^{x^2}}\\ &=e^{\lim_{x\to +\infty}(x-x^2\ln\left(1+\frac1x\right))}\\ &=e^{\lim_{x\to +\infty}x^2(\frac1x-\ln\left(1+\frac1x\right))}\\ &=e^{\lim_{x\to…
Shawn D
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Evaluating $\lim _{x \rightarrow 0} \frac{12-6 x^{2}-12 \cos x}{x^{4}}$

$$ \begin{align*} &\text { Let } \mathrm{x}=2 \mathrm{y} \quad \because x \rightarrow 0 \quad \therefore \mathrm{y} \rightarrow 0\\ &\therefore \lim _{x \rightarrow 0} \frac{12-6 x^{2}-12 \cos x}{x^{4}}\\ &=\lim _{y \rightarrow 0} \frac{12-6(2…
Roblox
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Existence of a limit or misspelled.

If someone proposes the problem: Calculate the limit: $\lim_{x\rightarrow 2}\frac{x-2}{2-\sqrt{4}}$ For me the limit does not exist because in fact the function $\frac{x-2}{2-\sqrt{4}}$ does not exist. However, it is also true that the problem is…
SSS1
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A question on limit

Suppose that $p_k>0, k=1,2,\dots, $ and $$ \lim_{n\to\infty} \frac{p_{n}}{p_1+p_2+\dots+p_n}=0,\quad \lim_{n\to\infty} a_n=a. $$ Show that $$ \lim_{n\to\infty}\frac{p_1a_n+p_2a_{n-1}+\dots+p_na_1}{p_1+p_2+\dots+p_n}=a $$ The hints are much…
Paul
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Show that $\lim\limits_{x\to0}\frac{e^x-1}{x}=1$

Show that $\displaystyle\lim_{x\to0}\frac{e^x-1}{x}=1$ Letting $y=e^x-1\implies e^x=y+1\implies x=\log(y+1)$ the evaluation is easy. But I can't understand how to express given function as a composition of two functions so that the following rule…
Sriti Mallick
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Show that $\lim \limits_{n\rightarrow\infty}\frac{n!}{(2n)!}=0$

I have to show that $\lim \limits_{n\rightarrow\infty}\frac{n!}{(2n)!}=0$ I am not sure if correct but i did it like this : $(2n)!=(2n)\cdot(2n-1)\cdot(2n-2)\cdot ...\cdot(2n-(n-1))\cdot (n!)$ so I have $$\displaystyle…
Devid
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Finding $\lim_{x \to 0+}\frac{x^{(\sin x)^x}-(\sin x)^{x^{\sin x}}}{x^3}$

Problem $$\lim_{x \to 0+}\frac{x^{(\sin x)^x}-(\sin x)^{x^{\sin x}}}{x^3}.$$ Attempt \begin{align*} &\lim_{x \to 0}\frac{x^{(\sin x)^x}-(\sin x)^{x^{\sin x}}}{x^3}\\ =&\lim_{x \to 0}\frac{\exp [(\sin x)^x\ln x]-\exp [{x^{\sin x}\ln(\sin…
mengdie1982
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