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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Solve limit with Lagrange theorem

I tried to solve this limit: $$ \lim_{x \to +\infty} x^2\left(e^{\frac{1}{x+1}}-e^{\frac{1}{x}}\right) $$ Instead of solving it with Taylor series (using $u = 1/x$), I noticed that the difference within the parenthesis is the $\Delta f$ of the…
claudia
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I need to find $\lim_{h \to 0, h\ne 0} \sqrt[h]{\frac{3^h+2^h}{2}}$

I need to find $\lim_{h \to 0, h\ne 0} \sqrt[h]{\frac{3^h+2^h}{2}}$. My attempt: $\lim_{h \to 0, h\ne 0} \sqrt[h]{\frac{3^h+2^h}{2}}=\lim_{h \to 0, h\ne 0}\exp(\frac{\log(\frac{2^h+3^h}{2})}{h})=\exp(\lim_{h \to 0, h\ne…
zesy
  • 2,565
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Calculating the limit $\lim_{x \to \infty } \left( \frac{a-1+b ^{ \frac{1}{x} } }{a} \right) ^{x}$ for $a>0, b>0$

If $a>0, b>0$ then what is the limit $$\lim_{x \to \infty } \left( \frac{a-1+b ^{ \frac{1}{x} } }{a} \right) ^{x}$$ I tried putting $y=\frac{1}{x}$ but it's not working.
mouse2
  • 33
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4 answers

is this correct $\lim_{ n \to \infty} \sum_{k=2^n}^{2^{n+1}} \frac{1}{k}= \ln 2$?

I think i might have seen this result somewhere $\lim_{ n \to \infty} \sum_{k=2^n}^{2^{n+1}} \frac{1}{k}= \ln 2$ but cant remember for sure. Is there a name for sums having limits in both lower and upper bound? is there anything similar for other…
jimjim
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The limit of part of a function, given the limit of the whole function

The question is: Suppose $\lim_{x\rightarrow 1}\frac{f(x)-7}{x-1}=4$, find $\lim_{x\rightarrow 1}f(x)$. The obvious answer is $7$, by going: $$\begin{align*} \lim_{x\rightarrow 1}\frac{f(x)-7}{x-1}&=4\\ \Rightarrow\lim_{x\rightarrow…
user303353
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How do I prove there is no limit at $(0,0)$?

$$f(x,y)=\frac{2^{xy}-1}{|x|+|y|}$$ I got to the conclusion that there is no limit, but I am not sure how to prove it.
Alon
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5 answers

Justifying the result of $\lim_{x\to\infty} \tfrac{3^x}{4^x}$

I am working with the next limit: $$\lim_{x\to\infty} \frac{3^x}{4^x}$$ I intuitively know that since $$3^x< 4^x$$ when $x$ tends to infinite, the result of the limit is: $$\lim_{x\to\infty} \frac{3^x}{4^x}=0$$ However, I need a some more…
Neo
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To prove the limit of $\frac{x^2+2\cos x-2}{x\sin^3 x}$ at zero is $1/12$

To Prove $$\lim_{x \to 0}\frac{x^2+2\cos x-2}{x\sin^3 x}=\frac{1}{12}$$ I tried with L'Hospital rule but in vain.
Vinay5forPrime
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Evaluate the limit $\mathop {\lim }\limits_{n \to \infty } \frac{{(n + 1){{\log }^2}(n + 1) - n{{\log }^2}n}}{{{{\log }^2}n}}$

Evaluate: $$\mathop {\lim }\limits_{n \to \infty } \frac{{(n + 1){{\log }^2}(n + 1) - n{{\log }^2}n}}{{{{\log }^2}n}}$$ Intuitively, I feel that for large $n$, ${\log}(n+1) \approx \ {\log}(n) $. So, the above limit should reduce to: $$=\mathop…
Kaustav Sen
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Limit of $\lim\limits_{n\to\infty}\frac{\sum_{m=0}^n (2m+1)^k}{n^{k+1}}$

I wanted to find the limit of: ($k \in N)$ $$\lim_{n \to \infty}{\frac{1^k+3^k+5^k+\cdots+(2n+1)^k}{n^{k+1}}}.$$ Stolz–Cesàro theorem could help but $\frac{a_n-a_{n-1}}{b_n-b_{n-1}}$ makes big mess here: $$\lim_{n \to…
Andy
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Proving Renyi entropy properties

I want to prove that Shannon entropy is a special case of Renyi entropy by solving this, $$ \lim_{\alpha\to1}\frac{1}{1-\alpha}\ln\sum_{k=1}^{n}p_{k}^{\alpha} = -\sum_{k}p_{k}\ln p_{k} $$
TBBT
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4 answers

Evaluating $\lim_\limits{x\to 1 }\bigl( (2^x x + 1)/(3^x x)\bigr)^{\tan(\pi x/2)}$

I have to calculate limit $$\lim_{x\to 1 } \left(\frac{2^x x + 1}{3^x x}\right)^{\tan(\frac{\pi x}{2})}.$$ I know $\tan(\frac{\pi x}{2})$ is undefined in $x = 1$, but can I just put $x = 1$ into $\frac{x\cdot 2^x + 1}{x\cdot3^x}$ and get…
Desh
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4 answers

The limit $\lim_{n\to \infty}{na^n}=0$

Let $a\in [0,1)$. I want to show that $$\lim_{n\to \infty}{na^n}=0$$ My try : $$na^n={n\over e^{-(\log{a})n}}$$ and the limit is $${+\infty\over +\infty}$$ Hence by l'Hopital's rule we have that $$\lim_{n\to \infty}{1\over…
palio
  • 11,064
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6 answers

How to compute this gross limit.

How do I compute this limit? $$ \lim_{n \to \infty} \frac{\left(1 + \frac{1}{n} + \frac{1}{n^2}\right)^n - \left(1 + \frac{1}{n} - \frac{1}{n^2}\right)^n }{ 2 \left(1 + \frac{1}{n} + \frac{1}{n^2}\right)^n - …
mick
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1 answer

Existence of a limit

Prove if the following is correct or not: If $$\lim _{x\to x_0}f(x) = L \text{ and } \lim_{x\to x_1}g(x) = x_0,$$ then $$\lim_{x\to x_1} f(g(x))= L.$$ So, I guess this can be solved either by proving it or find a example that contradicts the above,…
Belf
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