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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limit of functions $f(x)>g(x)$, what can we conclude about their limits

Let $f$ and $g$ be two functions defined on $S$ and $c$ be a cluster point of $S$. Suppose $f(x)\to l_1$ and $g(x)\to l_2$ as $x\to c$. If $f(x)>g(x)$ for any $x\in S\setminus{\{c\}}$,can we conclude $l_1 > l_2$? I think not and one of the counter…
Oneoneonesong
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Determining the limit: What does it mean when one obtains zero in the calculation?

Consider the sequence $a_n = \sqrt{n+\sqrt{n}}-\sqrt{n-\sqrt{n}}\\$. To determine the limit I did the following: \begin{aligned} a_{n} &=\left(\sqrt{n+\sqrt{n}}-\sqrt{n-\sqrt{n}}\right)…
Hilberto1
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$\lim_{n\to\infty}ne^{\frac{x}{n}}-n = x$?

Im fairly sure it's $x$. Here's what I've got so far. $\lim_{n\to\infty}ne^{\frac{x}{n}}-n = \lim_{n\to\infty}n(e^{\frac{x}{n}}-1) = \lim_{n\to\infty}n(-1+\sum_{i=0}^\infty\frac{x^i}{i!n^i}) =…
mathemagiker
  • 173
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What can I do this cos term to remove the divide by 0?

I was asked to help someone with this problem, and I don't really know the answer why. But I thought I'd still try. $$\lim_{t \to 10} \frac{t^2 - 100}{t+1} \cos\left( \frac{1}{10-t} \right)+ 100$$ The problem lies with the cos term. What can I do…
efox29
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If $n = xm$ and $n \rightarrow \infty$, then $m \rightarrow \infty$?

Did Stewart prove the result in green, himself or as an exercise, in Calculus Early Transcendentals or the normal version Calculus? I dislike duplicates as much as the next guy. A result like this, from first year calculus, must have gotten asked…
user851668
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Isn't $\lim\limits_{x \to 0}1^{1/x}$ be indeterminate

Isn't $\lim\limits_{x \to 0}1^{1/x}$ be indeterminate Since $1^{\infty}$ is indeterminate. Thanks
Rajesh Marndi
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Two variables function limit

In an exercise of a limit of a function of two variables in the solution I read this inequality: $$ \frac{x^2y^2}{(x^2+y^2)^\frac{3}{2}} \le \frac{1}{2}\sqrt{x^2+y^2} $$ how did they arrive at this result?
Salmon
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Question about limits with variable on exponent

So I have to find the following limit $$\lim_{n\to\infty}\left(1+\frac{2}{n}\right)^{1/n}.$$I said that this is $$\lim_{n\to\infty}\left[\left(1+\frac{2}{n}\right)^n\right]^{1/n^2}=\left(e^2\right)^{\lim_{n\to\infty}1/n^2}=1.$$Now I know that the…
Zugzwang14
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Evaluate $\lim_{n\to \infty} (n+1)^{\frac 23} -(n-1)^{\frac 23}$

Clearly, $\infty -\infty \not =0$ I have a feeling the squeeze theorem can be applied here, but I am not sure how to write the required terms separately. Can I get a hint?
Aditya
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How to find the limit of this function

We have the function $$\dfrac{\sqrt{n^4 + 100}}{4n}$$ I think the best method is by dividing by $n$, but I have no idea what that yields, mainly because of the square root.
LimitsMate
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Limit approximates $0.\infty$ to 0

A question defines $$f(x)=\begin{cases}g(x)\cos\frac{1}{x}&\text{if } x\neq0\\0&\text{if } x=0\end{cases}.$$ Here, $g$ is an even function and differentiable at $0$, and $g(0)=0$. You're required to find $f'(0)$. On differentiating $f(x)$ to find…
harry
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Finding the limit of a function defined by integral

I am trying (unsuccessfully) to find the following limit: $$ \lim_{t \to \infty} \frac{1}{t^2}\int_0^t\ln(2e^x+x^2)dx $$ Some background - supposedly, this exercise is meant to be similar to what we've done in class, where we found the limit: $$\lim…
Daniel Leza
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Game of sets. Limit points

Let $A, B \subset \mathbb{R}$. We know that $A' = B' = \varnothing$ and $(A + B)' = [0, 1]$, where $$A + B = \{a + b \; | \; a \in A, b \in B\}$$ Find $A, B $. I am trying to solve that problem for the second day in a row and the only thing that…
K-dizzled
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Is it valid to apply L'hopital rule to evaluate the limit?

$$\lim_{x\to 0^+}\frac{2\tan x(1-\cos x)}{\sqrt{x^2+x+1}-1}$$ In the book I am reading, the limit evaluated in this way: $$\lim_{x\to 0^+}\frac{2\tan x(1-\cos x)}{\sqrt{x^2+x+1}-1}\times \frac{\sqrt{x^2+x+1}+1}{\sqrt{x^2+x+1}+1}=\lim_{x\to…
Etemon
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$|f(x)| \leq g(x) \forall x$, and $\lim _{x \rightarrow a} g(x)=0$, what is $\lim _{x \rightarrow a} f(x)$? What if $\lim _{x \rightarrow a} g(x)=5$?

$$ \begin{aligned} &\text { Suppose }|f(x)| \leq g(x) \text { for all } x . \text { What can you conclude about } \lim _{x \rightarrow a} f(x) \text { if } \lim _{x \rightarrow a} g(x)=0 ?\\ &\text { What if } \lim _{x \rightarrow a} g(x)=5…
Jim Danger
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