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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Does $ \left\lceil \frac{1}{n}\right\rceil\to 0$ or $1$ as $n\to\infty ?$

I think the limit is $1$, because $\left\lceil \frac{1}{n}\right\rceil=1$ for all $n$, but it seems counter intuitive for some reason. If we crudely "replace $n$ with $\infty$" we get $\lceil 0\rceil=0$ (also, if it was $0$ I can't see a way of…
Little miss sunshine
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Constant sequence?

Consider a sequence with strictly positive terms $(a_n)_{n\geq1}$ with the property: $$\lim_{n\rightarrow \infty} \left(\frac{a_1}{a_2}+\frac{a_2}{a_3} + \cdots + \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}-n\right)=0$$ Prove that this sequence is…
medicu
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Wrong intuitive understanding of a limit?

In my textbook, before introducing the epsilon delta definition, they gave a working definition of what a limit is. The definition sounded something like this "$\lim \limits_{x \to a}f(x) = L$, if when $x$ gets closer to $a$, $f(x)$ gets closer to…
Carefullcars
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$\lim_{n\to\infty}|\sin n|^{\frac{1}{n}}$

One of my teachers have given a limit to compute: $$\displaystyle\lim_{n\to\infty}|\sin n|^{\frac{1}{n}}$$ I have proved that if the limit exits it has to be $1$. (By using the fact that $\{n\pi\}$ is dense in $[0,1]$) But I seem to have no idea…
Grobber
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Computing $\lim\limits_{n\to\infty}(1+1/n^2)^n$

Why is $\lim\limits_{n\to\infty}(1+\frac{1}{n^2})^n = 1$? Could someone elaborate on this? I know that $\lim\limits_{n\to\infty}(1+\frac{1}{n})^n = e$.
Bolz
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Finding out the limit $\lim_{a \to \infty} \frac{f(a)\ln a}{a}$

For any real number $a \geq 1$ let $f(a)$ denote the real solution of the equation $x(1+\ln x)=a$ then the question is to find out $$ \lim_{a \to \infty} \frac{f(a)\ln a}{a}$$. It is clear that if we denote $h(a)$ by $h(a)=a(1+\ln a)$ then $f(a)$…
Navin
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Finding the limit of a difficult exponential function

I am stumped on this question, any help or tips would be appreciated! Find the limit, if it exists: $$\lim_{x\to\infty } \left(\sqrt{e^{2x}+e^{x}}-\sqrt{e^{2x}-e^{x}}\right)$$
Jason Huang
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Limits of integrals

How would you show that if $f : [0,1] \rightarrow \Bbb R$ is continuous, then $$\lim_{n\rightarrow \infty}\int_0^1\int_0^1 \cdots \int_0^1 f\left( \frac{x_1+x_2+\cdots+x_n}{n} \right)~dx_1~dx_2\cdots dx_n = f\left( \frac{1}{2} \right)…
Emma
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Proving the limit

I have to prove that:$$\lim_\limits{x\to\infty}\bigg(\frac{n}{\frac{1}{x+a_1}+\frac{1}{x+a_2}+\cdots+\frac{1}{x+a_n}}-x\bigg)=\frac{a_1+a_2+\cdots+a_n}{n}$$ The way I started doing this…
A6SE
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How find $\lim\limits_{n \to \infty } \sqrt{n} \cdot \sqrt[n]{\left( \lim\limits_{n \to \infty } a_n\right)-a_n}$?

Let $a_n= \sqrt{1+ \sqrt{2+\cdots+ \sqrt{n} } }$ How find $\lim\limits_{n \to \infty } \sqrt{n} \cdot \sqrt[n]{\left( \lim\limits_{n \to \infty } a_n\right)-a_n}$ ?
piteer
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Finding $\lim\limits_{x \to 0}\ \frac{\sin(\cos(x))}{\sec(x)}$

The problem is to find: $\lim\limits_{x \to 0}\ \dfrac{\sin(\cos(x))}{\sec(x)}$ I rewrite the equation as follows: $\lim\limits_{x \to 0}\ \dfrac{\sin(\cos(x))}{\dfrac{1}{\cos(x)}}$ And multiply by $\dfrac{\cos(x)}{\cos(x)}$,…
waiwai933
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Without Taylor series $\lim\limits_{n\to\infty}\frac{n}{\ln \ln n}\cdot \left(\sqrt[n]{1+\frac{1}{2}+\cdots+\frac{1}{n}}-1\right) $

Is it possible to compute it without Taylor series? $$\lim_{n\to\infty}\frac{n}{\ln \ln n}\cdot \left(\sqrt[n]{1+\frac{1}{2}+\cdots+\frac{1}{n}}-1\right) $$ Maybe you try your luck without computational software.
Enlightened by God
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Limit involving binomial coefficients: $\lim_{n\to\infty}\left(\binom{n}{0}\binom{n}{1}\dots\binom{n}{n}\right)^{\frac{1}{n(n+1)}}$

I am facing difficulty with the following limit. $$ \lim_{n\to\infty}\left(\binom{n}{0}\binom{n}{1}\dots\binom{n}{n}\right)^{\frac{1}{n(n+1)}} $$ I tried to take log both sides but I could not simplify the resulting expression. Please help in this…
Navin
  • 2,605
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Why is it legit to evaluate $\lim_{x\rightarrow 1} \frac{(x-1)(x+1)}{x-1}$ by cancelling common factors?

I haven't ever taken an analysis course, so maybe that's where I would really learn this, but I've always wondered why it's okay to do this when evaluating a limit. I guess it's the case that there is a theorem which says that the limit of a…
crf
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Please explain how to solve limit. I know the answer but how to explain it?

Problem: $a@b = \frac{a+b}{ab+1}$. Solve limit: $\lim_{n \to \infty}(2@3@...@n)$. I've tried to solve this problem by just calculating: $$2 @ 3 = 0.714$$ $$2 @ 3 @ 4 = 1.222$$ $$2 @ 3 @ 4 @ 5 = 0.875$$ $$2 @ 3 @ 4 @ 5 @ 6 = 1.1$$ I found the…
Maximax67
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