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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How to find this limit as $x\to 0$?

$$\lim_{x\to 0}\frac{\sin(\tan x)-\tan(\sin x)}{\arcsin(\arctan x)-\arctan(\arcsin x)}$$ Hopital's rule and Taylor expansion are practically impossible. Is there a better way to do it (mathematica gives the answer $1$). This limit is of the form…
user5402
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Which theorem is it, or can somebody help with a proof to it?

$$\lim_{n\to \infty} a_n=0 \iff \lim_{n\to \infty} \frac{a_{n+1}}{a_n}<1 \text{ when } a_n>0 \text{ and } \frac{a_{n+1}}{a_n}>0.$$
Ilya
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Prove limit of $\lim\limits_{x\mapsto 0}x\cdot\sin\dfrac1x = 0$?

How do you prove $$\lim\limits_{x\mapsto 0}x\cdot\sin\dfrac1x = 0$$ with limit definition and without limit definition? My problem is that $\lim\limits_{x\mapsto 0}\sin\dfrac1x$ has not limit in zero so I can't use $$\lim\limits_{x\mapsto…
CVDE
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$\lim_{n \to \infty}n\sin\left(\pi/n\right)$

Assume unit radius. The area of regular polygon with $n$ sides = $n\sin\left(\pi/n\right)$, how does $n\sin\left(\pi/n\right)$ approach $1$ as $n \to \infty$ ?. $$ \mbox{I have tried}\quad \pi\,{\sin\left(\pi/n\right) \over \pi/n}\quad \mbox{so it…
W. Davis
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Tricky Limits Problem ()

Problem: If $\lim_{x \to 0}{\sin2x\over x^3}+a+{b\over x^2}=0$ then find the value of $3a+b$. My attempt: $\lim_{x \to 0}{\sin2x\over x^3}+a+{b\over x^2}=\lim_{x \to 0}{\sin2x\over 2x}({2\over x^2})+a+{b\over x^2}={2+b+ax^2\over x^2}$.From this we…
Student
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Evaluating $\lim \limits_{x \to 0} \frac {e^{-1/x^2}}{x}$ without using L' Hôpital's rule

Is there a way to evaluate $$\lim \limits_{x \to 0} \frac {e^{-1/x^2}}{x}$$ without applying L' Hôpital's rule? I've tried subbing $y = \frac {1}{x}$ and it did not work, and I also tried evaluating the expression by the exponent's Taylor's series…
Jacky Yip
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Limit $\lim_{x \rightarrow \infty, y \rightarrow \infty} \left( \frac{xy}{x^2 + y^2}\right)^{x^2} $

Given the followning limit: $$ \lim_{x \rightarrow \infty, y \rightarrow \infty} \left( \frac{xy}{x^2 + y^2}\right)^{x^2} $$ To find limit I have made following steps: Let $ x = y $ ,then limit equals $0$ Let $ x > y $ ,then consider the limit: $$…
eaniconer
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Why the the following limit is $1$? (This result is given by Maple.)

Why the the following limit is $1$? (This result is given by Maple.) $$\lim\limits_{x\rightarrow0}\,\frac{\ln(Ax+Bx^b)}{\ln(x)},$$ where $A>0,B>0,0
Dave
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How to evaluate a limit of the indeterminate form $(0/0)^0$

How to find the $\lim_{n \to \infty} \left(\dfrac{(n+1)(n+2)\cdots(n+2n)}{n^{2n}}\right)^{1/n}$? I know how to find it for the indeterminate form of $1^{\infty}$ by converting it into $0/0$ form, but this cannot be converted into any known…
Matt
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On finding a limit

So, my homework says: Find a value of $n \in\mathbb N$ from which there's certainty that: ... c) $\frac{(-1)^n}{n+1}+2$ is within 1.9 and 2.1 I read this as "make sure that the limit is $2$, within a margin of $2.1 - 1.9$", which, turns into…
Misguided
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to prove $e^x$ is continuous function using $\epsilon$ and $\delta$ method

How to prove $e^x$ is continuous We know that $$e^x=1+x+\frac{x^2}{2!}+...+\frac{x^n}{n!}+...$$ For all $|x|<1$ By algebra of continuous function we can prove $e^x$ is continuous .but but 1. how to prove it if $|x|>1$? Also 2.how to use…
Sathasivam K
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Getting the wrong result when solving a limit

I correctly solved to following limit like this: $$\lim_{x\to\infty} \left(\frac{1}{x^2}\right)^{\frac{2x}{x+1}}$$ $$ = \lim_{x\to\infty} (\frac{1}{x^2})^{\frac{2x}{x(1+\frac{1}{x})}}$$ $$ = \lim_{x\to\infty}…
Aaron
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What is my mistake when evaluating this limit?

$\lim_{x\to 0^+} \frac{1}{x}=\infty$ But with l'Hospital's Rule $\lim_{x\to 0^+} \frac{1}{x}=\lim_{x\to 0^+} \frac{0}{1}=0$ So where's my naive mistake?
Korgan Rivera
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What is $\lim\limits_{n\rightarrow\infty}\left(1-\left(1-\frac1n\right)^{f(n)}\right)^{2f(n)}$ when $f(n)$ grows faster than $n$?

What is $$\lim_{n\rightarrow\infty}\left(1-\left(1-\frac1n\right)^{f(n)}\right)^{2f(n)}$$ when $f(n)$ grows faster than $n$? Is the limit $1$? How fast should $f(n)$ grow for this to happen?
Turbo
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How to find limit $\lim \limits_{ x\rightarrow +\infty }{ \tan { { \left( \frac { \pi x }{ 2x+1 } \right) }^{ 1/x } } } $

I need to find this limit $\lim\limits_{ x\rightarrow +\infty }{ \tan { { \left( \frac { \pi x }{ 2x+1 } \right) }^{ 1/x } } } $. Give a hint please.Thanks
user315355
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