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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limit with nested power of $x$

Evaluation of $$\lim_{x\rightarrow 1}\frac{x^{x^{x^{x^{x^{x}}}}}-x^{x^{x^{x^x}}}}{(x-1)^6}$$ Try:: I try to solve it using Binomial Index for fraction $\displaystyle x^x=\bigg[1+(x-1)\bigg]^x=1+x(x-1)+\frac{x(x-1)^2}{2}+\cdots $ Did not know how…
DXT
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Name for multi-valued analogue of a limit

Let the "multilimit" $L_{x \to v}f(x)$ denote a set of values that $f(x)$ can approach as $x$ approaches $v \in \mathbb{R}$ . More formally, $$ \xi \in L_{x \to v}f(x) \stackrel{\text{def}}{\iff} \exists s \in \text{Seq}[\mathbb{R}] . \left(\lim_{t…
Greg Nisbet
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Is there a function $f: \mathbb{R} \to \mathbb{R}$ such that $\lim\limits_{x\to p}f(x)=\infty$ for every $p \in \mathbb{R}$

I was wondering if there is a function $f: \mathbb{R} \to \mathbb{R}$ such that its limit at every point is infinite. I guess not, because what would its graph look like, but then again, I don't know how to prove it.
user4201961
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$\displaystyle \lim_{x\to 0} \frac{e^{2x}-1}{3x} $ help please?

So I have to find the $$\displaystyle \lim_{x\to 0} \frac{e^{2x}-1}{3x} $$ I first solved this by L'hopital and got $\frac{2}{3}$ but now I read carefully and it says in my book that I shouldn't solve this by L'hopital..any hints?
rttrrt
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Limit of quotient

I recently learned that when you are solving for the limit of a quotient, you have to divide everything by the highest number in the denominator, like this $$ \lim_{x \to \infty} \frac{\sqrt{4 x^2 - 4}}{x+5} = \lim_{x \to \infty} \frac{\sqrt{4 -…
nox15
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Computation of limit: $\lim\limits_{n \to \infty} \frac{1}{n}\left ( \cos1+\cos(\frac{1}{2})+\cos(\frac{1}{3}) +...+\cos (\frac{1}{n}) \right )$

Help me please to compute this limit: $\lim\limits_{n \to \infty} \frac{1}{n}\left ( \cos1+\cos(\frac{1}{2})+\cos(\frac{1}{3}) +...+\cos (\frac{1}{n}) \right )$ Thank you.
Panka
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Find the limit of $(\tan(x) + \sec (x))^{1/\sin(x)}$

$$ \lim_{x \rightarrow 0} [\tan(x) + \sec(x)]^{\csc(x)} = e $$ how to arrive at e, according to wolfram alpha, that this is the answer?
user29418
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$\lim_{n\rightarrow \infty}\left[\frac{\left(1+\frac{1}{n^2}\right)\cdot \cdots\cdots \left(1+\frac{n}{n^2}\right)}{\sqrt{e}}\right]^n$

$$\lim_{n\rightarrow \infty}\Bigg[\frac{\bigg(1+\frac{1}{n^2}\bigg)\bigg(1+\frac{2}{n^2}\bigg)\cdots\cdots \bigg(1+\frac{n}{n^2}\bigg)}{\sqrt{e}}\Bigg]^n$$ Try: $$y=\lim_{n\rightarrow…
DXT
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Understanding when it's OK to use limits arithmetic of multiplying

I'm having hard time understanding when I can do the following move: $\lim_{n\to \infty}a_n\cdot b_n = \lim_{n\to \infty}a_n\cdot\lim_{n\to \infty}b_n $ On one hand, my teacher said that I can do that only if prove that both limits of $a_n$ and…
Avishay28
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Find $\lim\limits_{(x,y)\to (0,0)}{\frac{e^{x}+y-1}{x+y}}$

Find $$\lim_{(x,y)\to (0,0)}{\dfrac{e^{x}+y-1}{x+y}}$$ I tried with different trajectories and I always get that the limit is $1$, but I cannot prove it, any help?
Nah
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is $0.\overline{99}$ the same as $\lim_{x \to 1} x$?

So we had an interesting discussion the other day about 0.999... repeated to infinity, actually being equal to one. I understand the proof, but I'm wondering then if you had the function... $$ f(x) = x* \frac{(x-1)}{(x-1)} $$ so $$ f(1) =…
tbischel
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Suppose that $\lim_{x\to \infty} f'(x) = a$. Is it true that $\lim_{x\to \infty} {f(x)\over x} = a$

Suppose that $\lim_{x\to \infty} f'(x) = a$. Is it true that $\lim_{x\to \infty} {f(x)\over x} = a$ If so, can you prove it? Thanks!
pbk83
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Can I break this limit into individual terms?

$$\lim_{x\to \infty} {\frac{x}{x^2+1} +\frac{x}{x^2+2} + ... + \frac{x}{x^2+x} }$$ It seems obvious that the result is zero for each term but in order to break the limit into its individual parts we must know that every term's limit exists .
Χριστίνα Κ.
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Limit with sin indeterminate

How do I calculate the following limit $$\lim_{x\to\infty} \frac{3x-\sin x}{x+\sin x}$$ It's an indeterminate limit but how can I solve it? Does it help if I split it?The answer I got is $-1$ but it's $3$.
Lola
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Limit with $\tan$ L'Hopital

I have kind of a simple and maybe stupid question but $$ \lim_{x\to 0} \frac{(\tan x-x)}{x^3} $$ why am I not allowed to split the limit like this : $$\lim_{x\to 0} \frac {\tan x}{x^3} -\frac{x}{x^3}$$ which equals $0$? I came up with the right…
Lola
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