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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Calculating $\lim\limits_{x \to -\infty}(1+\tan(-\frac{4}{x}))^\frac{1}{\arctan \frac{3}{x}}$

Can you help me with solving this limit? $\lim\limits_{x \to -\infty}(1+\tan(-\frac{4}{x}))^\frac{1}{\arctan \frac{3}{x}}$ Thank you. Edit: Is there any solution without L'Hospital?
heky__
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Limit of $n!^{1/n^2}$ using Squeeze Theorem?

I'm trying to find the limit of $n!^{1/n^2}$ as $n$ goes to infinity. What I've done so far is: I know that $n < n^2 < n! < n^n$ for large $n$, and I know that $n^{1/n^2} = 1$, but I'm not sure how to find the limit of $n^{n^{1/n^2}}$.
user270494
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If $ \lim _{x\rightarrow -\infty }\sqrt {x^{2}+6x+3}+ax+b=1 $ then find $a+b$

$ \lim _{x\rightarrow -\infty }\sqrt {x^{2}+6x+3}+ax+b=1 $ if I use this $$\lim _{x\rightarrow -\infty }\sqrt {ax^{2}+bx+c}=\left| x+\dfrac {b} {2a}\right| $$ I find $a=1,b=4$ but if I try to multiple by its conjugate $$\lim _{x\rightarrow -\infty…
memonto
  • 321
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Prove if $\lim\limits_{n\rightarrow\infty}(a_1+a_2+...+a_n)=S$, then $\lim\limits_{n\rightarrow\infty}\frac{a_1+2a_2+...+na_n}{n}=0$

This is a homework question Prove if $\lim\limits_{n\rightarrow\infty}(a_1+a_2+...+a_n)=S$, then $\lim\limits_{n\rightarrow\infty}\frac{a_1+2a_2+...+na_n}{n}=0$ And this is my try to prove it, Let $k < n$,…
0xBBC
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Prove that $x^m/e^x\to0$ when $x\to+\infty$, for every $m$

Defn: We write $\lim_{x \uparrow \infty} f(x)=L$ whenever $f$ is defined on some unbounded interval such as $00$. $\exists x_0$ such that $\mid f(x) - L \mid < \epsilon$ whenever $x>x_0$ Prove that for…
llawliet_78
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How to calculate the limit of this function?

I wish to find the limit of this function as $x \rightarrow \infty$. $$f(x)=\left\{\begin{matrix} 1-\frac{1}{x} ~:\forall x \in \mathbb{Q}\\1 ~:\forall x \notin \mathbb{Q} \end{matrix}\right.$$ I have never had to find the limit of a function…
Samuel
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How to find the limit of a function?

Find $$\lim\limits_{n \to \infty} \left( \frac{n^{1/3}}{2} \text{arccos} \left(\dfrac1{\sqrt{1+\frac{4}{(k(n)-1)^2}}\sqrt{1+\frac{8}{(k(n)-1)^2}}} \right) \right).$$ where $k(n) = \dfrac{1}{12} (108n+12 \sqrt{768+81n^2})^{1/3}-\dfrac{4}{ (108n+12…
Sh.N.
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Proving $\lim_{n\to \infty} \frac{(\ln n)^a}{n^b} = 0$ for all $a,b>0$

How can I prove this: $$\lim_{n\to \infty} \frac{(\ln n)^a}{n^b} = 0 \quad \forall\, a,b > 0$$ Any ideas or tips? I tried to use L'Hôpital's rule but that led into nothing.
今天春天
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Problem with limit of recursive sequence having the actual index "n" in denominator.

The given sequence is: $$ a_1 = 1; \quad a_{n+1} = \frac{2a_n}{n+1} $$ $n$ is natural. What is the limit if $n\to\infty$? Please help!
Próba
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How to find limit of function: $\lim\limits_{n \to \infty} \frac{\sqrt{n}}{2} \arccos(\frac{n-2}{22+n})$

How would I find this limit? $\lim_{n \to \infty} \frac{\sqrt{n}}{2} \bigl(\arccos(\frac{n-2}{22+n}))$
Sh.N.
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what is the limit?

$$\lim_{x\to0+} (\sin x)^{2/\ln x}$$ Not too sure what method to use. I have tried to follow all the helpful answers and this is my working I have done. Please tell me if I have made a mistake. Many thanks! is this answer correct?
user212645
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$f'(x) > 0$, does that imply the limit as $x\rightarrow \infty$ is $\infty$?

Given the following situation: $f'(x) > 0$ and $f: \mathbb{R} \rightarrow \mathbb{R}$. I am trying to find if there is a flaw in my understanding of the properties of this situation. I assert that in the situation where $x$ approaches $\infty$ that…
user184881
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Let $x_0=2\cos\frac{\pi}{6}$ and $x_n=\sqrt{2+x_{n-1}},n=1,2,3...,$ prove that $\lim_{n\to \infty}2^{n+1}\sqrt{2-x_n}=\frac{\pi}{3}$

Let $x_0=2\cos\frac{\pi}{6}$ and $x_n=\sqrt{2+x_{n-1}},n=1,2,3...,$ prove that $\lim_{n\to \infty}2^{n+1}\sqrt{2-x_n}=\frac{\pi}{3}$ I am not able to correctly solve it,made some attempts,but no luck.This $2^{n+1}$ is creating problem.How should i…
user1442
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$\lim_{x\to a}\frac{1}{(x^2-a^2)^2}\left(\frac{a^2+x^2}{ax}-2\sin\left(\frac{a\pi}{2}\right)\sin\left(\frac{\pi x}{2}\right)\right)$

Prove that $\lim_{x\to a}\frac{1}{(x^2-a^2)^2}\left(\frac{a^2+x^2}{ax}-2\sin\left(\frac{a\pi}{2}\right)\sin\left(\frac{\pi x}{2}\right)\right)=\frac{\pi^2 a^2+4}{16a^4}$ where $a$ is an odd integer. I tried to apply L Hospital rule in this question…
user1442
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Is the limit $\lim_{x\rightarrow 0} \frac{1/(x+3)-1/3}{x}$ equal to $-\frac{1}{9}$?

$$\lim_{x\rightarrow 0} \frac{\frac{1}{x+3}-\frac{1}{3}}{x}$$ Is the limit $-\displaystyle{\frac{1}{9}}$?
FoolishNumber
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