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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Is $1/(x^2 + y^2)$ continuous?

I'm trying to check whether the function $$ f(x,y)=\begin{cases} \dfrac{1}{x^2+y^2} & \text{for $(x,y)\ne(0,0)$}\\[6px] 0 & \text{for $(x,y)=(0,0)$} \end{cases} $$ is continuous. My problem is with trying to check if…
Pedro Destri
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Trigonometric and exp limit

Evaluation of $$\lim_{x\rightarrow \frac{\pi}{2}}\frac{\sin x-(\sin x)^{\sin x}}{1-\sin x+\ln (\sin x)}$$ Without Using L hopital Rule and series expansion. $\bf{My\; Try::}$ I have solved it using L hopital Rule and series expansion. But I did…
juantheron
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$\lim_{x\to 0}(1+\sin(x)-x)^{1/x^3}$

Evaluate$$\lim_{x\to 0}\left(1+\sin(x)-x\right)^{{1}/{x^3}}$$ i was able to solve it using a taylor exp' for $\sin(x)$ but id like to know if there is a "simpler " way. something along the lines of $e^{\log}$-ing it, or L'hopital-ing it...
Rubenz
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Find the value of limit satisfying the given condition

Let $f(x)= \cos(x/2)\cdot\cos(x/4)\cdots\cos(x/2^n)$.If lim $ \lim_{n \to infinity} f(x) =g(x)$ , such that $\lim_{x \to 0} g(x)=k$, then the value of $\lim_{k \to 1} (1-k^{2011})/(1-k)$ I want to know that my solution is correct or not. If it…
Aakash Kumar
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Determine if $a_n=\frac{\ln^2(n)}{n^{1/2}}$ is convergent by using the properties of limits

Determine if $\lim\limits_{n \to \infty} a_n$=$\frac{\ln^2(n)}{\sqrt{n}}$ is convergent by using the properties of limits So I will take the limits of both top and bottom: $\lim\limits_{n \to \infty}\ln^2(n))=\infty$ and $\lim\limits_{n \to…
stackdsewew
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Squeeze Principle and Sin x

How does the Squeeze Principle used to find that $\lim_{x \rightarrow \infty} \sin x / x$? I understand what the squeeze principle is, but do not understand how to use it for this example.
Romeo
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Limit with root and fraction

I have this limit: $$\lim_{x\to 0^+} \frac{\sqrt{1+2x} - e^x}{x\arctan{x}}$$ And I try this \begin{align}\lim_{x\to 0^+} \frac{\sqrt{1+2x} - e^x}{x\arctan{x}} & = \lim_{x\to 0^+} \frac{\sqrt{1+2x} - e^x}{x\arctan{x}}\frac{\sqrt{1+2x} +…
user345990
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Conditions under which function is continuous

I have the following graph: $$h(t) = \begin{cases} 2t+1, & \mathrm{if}\ t \le -1, \\ 3t, & \mathrm{if}\ -1 < t < 1, \\ 2t-1, & \mathrm{if}\ t \ge 1.\end{cases}$$ The question I have to answer is: Give the conditions which would make the function $h$…
Michael Frey
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How to Solve $\lim_{x\to \infty}\sqrt{x^2 + ax} - \sqrt{x^2 + bx}$

I can't quite figure out how to manipulate this into a determinate form — should I try rationalizing it by multiplying by the conjugate, completing the square, or something like that? Note: I'm in precalc, so haven't learned any fancy calculus…
James
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Evaluation of product of $n$ terms of series , If $n\rightarrow \infty$

If $a_{1},a_{2},a_{3},.....,a_{n}$ are n terms of series such that $$\frac{n+1}{a_{n+1}}-\frac{n-1}{a_{n}} = \frac{2(n+2)}{n}\;,n\geq 1, n\in \mathbb{N}$$ Then $\displaystyle n^4\lim_{n\rightarrow \infty}\prod^{n}_{r=1}a_{r} = $ $\bf{My\; Try::}$…
juantheron
  • 53,015
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Help with finding limit of function with 2 variables

Find $$\lim_{(x,y)\to(0,0)} \frac{e^{\sin(x^2+y^2)}-1}{x^2+y^2}$$ I have tried taking limits when $y=x$ and when $y=-x$ but failing to get close to an answer.
Andre Croucher
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limit to infinity : trouble with l'hopital

Given the following limit for s positive constant $\lim_{x\to \infty} xe^{-sx}(\sin x-s\cos x) $ how can I prove that the above is equal to $0$ ? I re-write the limit as $ \frac{x(\sin x-s\cos x)}{e^{sx}} $ and then I use de l'Hopital theorem…
kaithkolesidou
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Proof of a limit problem

Let $C_{n}(u,v)=v-[max\{(1-u)^{1/n}+v^{1/n}-1, 0\}]^{n}$ under the constraints that $0 \leq u \leq 1, 0 \leq v \leq 1$. Prove that, $$\lim_{n\to\infty} C_{n}(u,v)=uv$$ Thanks in advance.
Dora Sengupta
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does $\lim_{N\to\infty}\frac{\sum_{i=1}^N a_i}{\sum_{i=1}^N b_i}$ converge to $\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^N\frac{a_i}{b_i}$

Can this ever be the case? $$\lim\limits_{N\to\infty}\frac{\sum\limits_{i=1}^N a_i}{\sum\limits_{i=1}^N b_i} = \lim\limits_{N\to\infty}\frac{1}{N}\sum_{i=1}^N\frac{a_i}{b_i}$$ with $a_i>0$, $b_i>0$, $a_i
user330617
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for $f(x)=\ln({1-\frac{1}{x^2})}$ find $\lim_{n\to\infty}{f(1)+f(2)+...+f(n)}$

Let $f:(1, \infty)\to \Bbb R$ with $f(x)=\ln\left({1-\frac{1}{x^2}}\right)$ Find $$\lim\limits_{n\to\infty}\left[f(1)+f(2)+...+f(n)\right]$$ What I have done so far is $$\begin{align} S_n&=\sum_{k=2}^n{f(k)} \\ &= \ln\left(\prod_{k=2}^n…
oren revenge
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