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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Find the Limit $\lim_{n \rightarrow \infty}\frac{1}{(n+1) \log (1+\frac{1}{n})}$

Find the limit $$\lim\limits_{n \rightarrow \infty}\frac{1}{(n+1) \log (1+\frac{1}{n})}$$
El Chapo
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4 answers

Simple limit problem with $a^x$.

Any idea of calculating the limit of $f(x)=2^x/3^{x^2}$ when $x$ approaches infinate? I looked up the function in geogebra and the limit is zero. Not sure how to prove it though... I guess using the L' Hospital (not sure if spelled correctly) rule.
user253724
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An interesting limit of certain $q$-binomial sums.

Let $\binom{n}{j}_q$ be a $q$-binomial coefficient. I would like to find a simple method to prove that $$ \lim_{q\to1}\frac{\sum\limits_{j = 0}^{2n} (-1)^j q^{m(j^2+j)} \binom{2n}{j}_q}{\sum\limits_{j = 0}^{2n} (-1)^j q^{(j^2+j)}…
Johann Cigler
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How to evaluate $\frac{2^{f(\tan x)}-2^{f(\sin x)}}{x^{2}f(\sin x)}$ as $x \to 0$?

If $f(x+y)=f(x)+f(y)$ for all real values of $x,y$. Given $f(1)=1$ How to evaluate $$\lim_{x \to 0} \frac{2^{f(\tan x)}-2^{f(\sin x)}}{x^{2}f(\sin x)}$$
user220382
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How to evaluate $\lim_{n\to\infty} a_n$

If $a_1=1$ and $a_{n+1}=\frac{4+3a_n}{3+2a_n}$,$n\geq1$, then how to prove that $a_{n+2}>a_{n+1}$ and if $a_n$ has a limit as ${n\to\infty}$ then how to evaluate $\lim_{n\to\infty} a_n$ ?
user220382
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Finding $\lim_{x\to\infty}\left(\frac{a^x-1}{x(a-1)}\right)^{1/x}$

Find the limit $\displaystyle\lim_{x\to\infty}\left(\dfrac{a^x-1}{x(a-1)}\right)^{1/x}$. While solving limits from my book I found this in the exercise. It says the answer is $1$ for $a<1$ and $a$ for $a>1$. I have tried different method like…
Grobber
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Obtaining $\nu \log (1+\frac{\theta}{\alpha}) $ as limit of $(\delta \gamma)- \delta (\gamma^{1/\kappa} + 2 \theta)^\kappa$

Can I obtain the expression $$ \nu \log (1+\frac{\theta}{\alpha}) $$ as a limit from the expression $$ \delta \gamma- \delta (\gamma^{1/\kappa} + 2 \theta)^\kappa $$ with $\kappa \to 0$ where $\nu$ and $\alpha$ can be chosen dependent of $\delta,…
htd
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If $0^0 = 1$ what is $\lim_{x\rightarrow 0} 0^{x}?$

If $0^0 = 1$ what is $\lim_{x \rightarrow 0} 0^{x}?$ Intuitively, it would appear to be equal to 1 as well since $\lim_{x \rightarrow \infty} f(x) = x = 0$. At the same time, if I consider a member of the set of points approaching the left-sided…
letsmakemuffinstogether
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Can anyone provide me a hint for finding the limit involving a factorial function?

I have to find the limit of this function: $\lim _{n\rightarrow \infty }\frac{(n!)^{2}}{(2n)!}$ Can anyone provide a hint?
bpr3003
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Find $\lim_{x\to 0}\frac{\sin x-x\cos x}{x^3}$

$$\lim_{x\to 0}\frac{\sin x-x\cos x}{x^3}$$ How do I go about doing this? I can see no simple way of doing this. Application of l'Hopital's rule would be very laborious. A Taylor expansion seems feasible but is that the best way? It seems like it…
RobChem
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Limit as $x$ tends to zero of $\frac{\csc(x)}{x^3} - \frac{\sinh(x)}{x^5}$

How would I find the $$\lim_{x\to0}\left(\frac {\csc(x)}{x^3} - \frac{\sinh(x)}{x^5}\right)$$ The only way I know how to do this is with l'hopitals rule but I don't see it helping here as we have x's in our denominator.
Goods
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limit $x$ tends to zero $(f(x^2) - f(x)) / (f(x)-f(0))$

If $f$ is differentiable and a strictly increasing function, then what is the following? $$\lim_{x\to0}\frac{f(x^2) - f(x)}{f(x)-f(0)}$$ Thank you.
zibzabpuzzless
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Sufficiently rigorous limit proof?

I needed to make the substitution $x = \sqrt{t^2+1} - t$ in an integral where in one limit $t\to+\infty$. So $x\to 0$ at this limit. I am aware you could complete the square under radical with an expression greater than $\sqrt{t^2+1} - t$, and then…
user140591
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Exercise of limit function definition

Suppose the function $f :\mathbb{R}\to\mathbb{R}$ has limit $L$ at $0$, and let $a > 0$. If $g :\mathbb{R}\to\mathbb{R}$ is defined by $g(x) := f(ax)$ for $x\in \mathbb{R}$. Show that $\displaystyle{\lim_{x\to 0} g(x) = L}$.
Roiner Segura Cubero
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How to show $\sum_{x=0}^{\infty }{\frac{a^{x}}{x!}\; =\; e^{a}}$

I've got a basic problem here with deriving the poisson distribution, where in part this summation is needed to show the expectation of the distribution is a parameter of the distribution's function, but while I can see from the derivation that this…
Topher
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