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 $\lim_{x \to \infty} \left(\frac{2x-3}{2x+5}\right)^{2x+1}$ using L'Hospital's Rule

How to find $$\lim_{x \to \infty} \left(\frac{2x-3}{2x+5}\right)^{2x+1}$$ When I am calculating the limit I get a form like $\infty \times \infty$. I can't continue from that point. I made it as $\frac{\infty}0$. But L'Hospital's Rule can't apply…
user6332995
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$\lim_{x\to a^{-}} f(x) =\lim_{x\to a^{+}} f(x) =L$

We know that : $$\lim_{x\to a} f(x) =L$$ if and only if : $$\lim_{x\to a^{-}} f(x) =\lim_{x\to a^{+}} f(x) =L$$ now : let $$f(x)=\sqrt{x}\\\lim_{x\to 0^{+}} \sqrt{x} =0\\\lim_{x\to 0^{-}} \sqrt{x} =!!!$$ so : $$\lim_{x\to 0} \sqrt{x}= \text{Does not…
user402750
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Find $\lim_{ n \to \infty }\left( (\frac{1}{n})^{17}+(\frac{2}{n})^{17}+(\frac{3}{n})^{17}+...+(\frac{n}{n})^{17} \right) =?$

Find the limit : $$\lim_{ n \to \infty }\left( (\frac{1}{n})^{17}+(\frac{2}{n})^{17}+(\frac{3}{n})^{17}+...+(\frac{n}{n})^{17} \right) =?$$ My Try: $$\lim_{ n \to \infty }…
Almot1960
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Is this an alright proof of an $e$ limit?

My goal is to prove that $$f(a) = \lim_{w \to 0} (1+aw)^{\frac{1}{w}} = e^a$$ without being too rigorous (just rigorous enough to convince myself that it really is true). Is the following method alright; or does it have flawed logic?…
QCD_IS_GOOD
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$\lim _{x\rightarrow\infty} \left(\sqrt{(1+ab)(1+ab+(1-a)cx^{-d})}-\sqrt{ab(ab+(1-a)cx^{-d})}\right)^{-x}$

I am interested in the limit as $x\rightarrow\infty$ of the following function: $$f(x)=\left(\sqrt{(1+ab)(1+ab+(1-a)cx^{-d})}-\sqrt{ab(ab+(1-a)cx^{-d})}\right)^{-x}$$ Here $0
M.B.M.
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How to evaluate this limit $\lim_{x\rightarrow\infty}(1+\sin(x))^{x}$

How to evaluate this limit $\lim_{x\rightarrow\infty}(1+\sin(x))^{x}$ I try in many forms but I cannot evaluate this limit; please help with tips please.
rcoder
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Evaluating limit $\lim_{k\to \infty}\prod_{r=1}^k\cos{\left(\frac {x}{2^r}\right)}$

I stumbled across the following question which asked to evaluate... $$\lim_{k\to \infty}\prod_{r=1}^k\cos{\left(\frac {x}{2^r}\right)}$$ I at first tried writing few terms $$\cos{\left(\frac {x}{2}\right)}\cos{\left(\frac…
LM2357
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Compute $\lim_\limits{x\to0^+}\frac{\pi/2- \arctan(1/x^2)-\sin(x^3)-1+\cos(x)}{x\tan(x)+e^{x^3}-1}$

I have to compute $$ \lim_{x\to0^+}\frac{\pi/2- \arctan(1/x^2)-\sin(x^3)-1+\cos(x)}{x\tan(x)+e^{x^3}-1} $$ I separated the numerator so I got that $$\dfrac{-1+\cos(x)}{x\tan(x)+e^{x^3}-1} \longrightarrow -\dfrac{1}2;$$ I know that the limit is…
Lorenzo B.
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$\lim_{x\to2}\frac{\sqrt{1+\sqrt{x+2}-\sqrt3}}{x-2}$

$\lim_{x\to2}\frac{\sqrt{1+\sqrt{x+2}-\sqrt3}}{x-2}$ My…
Brahmagupta
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Limit of $(1/n)^{(1/n!)}$ as $n \to \infty$

A computer algebra system told me that \begin{equation} \lim_{n \to \infty} \left( \frac{1}{n} \right)^{1/n!} = 1 \end{equation} How can I show this? I tried applying the exponential and logarithm to see that this is equal to \begin{equation} \exp…
Joshua Ruiter
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Limit of a sequence by Cauchy second test

The sequence is $$ \left[ \bigg(1+\frac{1}{n}\bigg)\bigg(1+\frac{2}{n}\bigg)\bigg(1+\frac{3}{n}\bigg)\cdots\bigg(1+\frac{n}{n}\bigg) \right]^{1/n} $$ as $n$ goes to infinity. By Cauchy second test it's pretty clear that it's limit will be equal to…
Nitish
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Why the definition of limit have a strict inequality rather than not?

My question is why the definition of limit of sequence is for $\epsilon >0$, there exist $N$ st for $n\ge N$ that $|x_n-L|<\epsilon$ instead of $|x_n-L|\le \epsilon$
Mathematics
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$\lim_{n\to\infty}n\sin(2^n \pi \sqrt{e}\mathrm n!)=?$

We know that $$\lim_{n\to\infty}n\sin(2\pi \mathrm en!)=2\pi$$ now : $$\lim_{n\to\infty}n\sin(2^n \pi \sqrt{e}\mathrm n!)=?$$ I tried:
Almot1960
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Incorrect solution in limits

$$ \lim_{n\to \infty}\left(\frac{1}{n^4}{+3^{\frac{2}{2+n}}}\right)^{n} $$ So i re-write it like: $\lim_{n\to \infty}e^{n\ln{\frac{1}{n^4}\ln3^{\frac{2}{2+n}}}}$ $=$ $e^{\frac{2n}{2+n}\ln{\frac{1}{n^4}\ln3}}=e^{{2}\ln{\frac{1}{n^4}\ln3}}$ So here,…
Музаффар Шакаров
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Need a tip/hint evaluating a limit

I have the following limit: $$\lim_{x\rightarrow\infty}\left(1+\frac{a}{x^{1/2+\epsilon}}\left(1-\exp\left(-\frac{b}{x^{1/2+\epsilon}}\right)\right)\ln\left(\frac{a}{x^{1/2+\epsilon}}\right)\right)^x$$ where $0
M.B.M.
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