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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Solving $\lim\limits_{n\to\infty}\frac{n^n}{e^nn!}$

I was solving a convergence of a series and this limit popped up: $$\lim\limits_{n\to\infty}\frac{n^n}{e^nn!}$$ I needed this limit to be $0$ and it is in fact (according to WolframAlpha), but I just don't see how to get the result.
GorTeX
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How to evaluate $\lim\limits_{p \rightarrow \infty} \left(\sum_\limits{i=1}^n \left|x_i-y_i\right|^p\right)^{\frac{1}{p}}$

I'd like to know why $\lim\limits_{p \rightarrow \infty} \left(\sum_\limits{i=1}^n \left|x_i-y_i\right|^p\right)^{\frac{1}{p}} = \max\limits_{1\le i \le n} \left| x_i-y_i\right|$ for $\mathbf{x},\mathbf{y}\in \mathbb{R}^n$. So I started by checking…
mauna
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limit of $ \lim\limits_{x \to ∞} \frac {x(x+1)^{x+1}}{(x+2)^{x+2}} $

Hello I am trying to find the limit of $ \lim\limits_{x \to ∞} \frac {x(x+1)^{x+1}}{(x+2)^{x+2}} $ I've tried applying L'H rule but it ends up getting really messy. The answer is $ \frac {1}{e} $ so I assume it must simplify into something which I…
Pyrons
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What is $\lim_{n\rightarrow \infty} \frac{1}{n^2}(\ln(\frac{2^n}{3^n})+\ln(\frac{5^n}{4^n})+\cdots+\ln(\frac{(3n-1)^n}{(n+2)^n}))$?

Per the title of this question, how does one go about calculating $$\lim_{n\rightarrow \infty} \frac{1}{n^2}\left(\ln\left(\frac{2^n}{3^n}\right)+\ln\left(\frac{5^n}{4^n}\right)+\cdots+\ln\left(\frac{(3n-1)^n}{(n+2)^n}\right)\right)\ ?$$ Thanks!
josh
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Elementary way to show $\lim_{n \rightarrow \infty} \sqrt[n]{a_n} = \lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n}$?

Let $a_n \gt 0$ for $n \in \mathbb{N}$. The convergence radius of the series $\sum_{n=0}^\infty a_n z^n$ is $\frac{1}{q}$ with $q = \lim_{n \rightarrow \infty} \sqrt[n]{a_n}$ or $q = \lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n}$, if these limits…
Keba
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Prove $\lim_{x \to 0} \frac{e^{\sin(x)} - e^{\tan (x)}}{e^{\sin (2x)}-e^{\tan (2x)}} = \frac{1}{8}$

Here's a nice little problem. $$\lim_{x \to 0} \frac{e^{\sin(x)} - e^{\tan (x)}}{e^{\sin (2x)}-e^{\tan (2x)}}$$ What's the quickest way to do this? One line solutions will be applauded :D Cheers, my jolly people :D
Nick
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Hypothetical contradiction to Bolzano-Weierstrass

we've learned about the Bolzano-Weierstrass theorem that states that if a sequence is bounded, then it has a subsequence that converges to a finite limit. Let's define $a_n$ as the digits of $\pi$, i.e. $a_1$ = 3, $a_2$ = 1, $a_3$ = 4, and so on…
blz
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How can I evaluate $\lim_{n \to\infty}\left(1\cdot2\cdot3+2\cdot3\cdot4+\dots+n(n+1)(n+2)\right)/\left(1^2+2^2+3^2+\dots+n^2\right)^2$?

How can I evaluate this limit? Give me a hint, please. $$\lim_{n \to\infty}\frac{1\cdot2\cdot3+2\cdot3\cdot4+\dots+n(n+1)(n+2)}{\left(1^2+2^2+3^2+\dots+n^2\right)^2}$$
Anna
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Solve limit using the concept of equivalent functions

How to solve this limit $$\lim_{x\rightarrow 0}{\frac{(2+x)^x-2^x}{x^2}}$$ using the concept of equivalent functions? For example, if $x\rightarrow 0 $ function $\sin x$ is equivalent to $x$, $\ln(1+x)\sim x$, $a^x-1 \sim x \ln a$, etc.
Alexander Side
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Proof for $\lim_{n \to \infty}\left(1+\frac{a_n}{n}\right)^n = e^a$?

In Statistical Inference by George Casella, Lemma 2.3.14 states that: $\text{Let }a_1,a_2,...\text{be a sequence of numbers converging to }a\text{, that is, }\lim_{n \to \infty}a_n=a\text{. Then}$ $$\lim_{n \to \infty}\left(1+\frac{a_n}{n}\right)^n…
C.J. Jackson
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Limit of a Sequence involving 3rd root

I'm not finding any way to simplify and solve the following limit: $$ \lim_{n \to \infty} \sqrt{n^2+n+1}-\sqrt[3]{n^3+n^2+n+1} $$ I've tried multiplying by the conjugate, but this give a more complex limit.
michalbures
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How to solve this limit $\lim_{n\to\infty} ((\frac{1}{\sqrt{n^2+1}}) + \cdots + (\frac{1}{\sqrt{n^2+n}}))$

How do I solve this limit $$\lim_{n\to\infty} \left(\frac{1}{\sqrt{n^2+1}} + \cdots + \frac{1}{\sqrt{n^2+n}}\right)\text{ ?}$$ (n goes to plus infinity.) I tried putting in $n=1,2,3,4,\ldots$ to find some pattern but it's hard to see where it's…
bodacydo
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Prove that the limit does not go to 6

I want to prove that $$ \lim_{x \to 2} \ x + 3 \ne 6 $$ What I thought about doing was first assuming the limit actually equaled $6$. Then taking an $x$ below and above $3$ and then finding a contradiction form the two statements 1) choosing $x =…
Treesrule14
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Statement equivalent to Limit of sequence

Def: A sequence $X = (x_n)$ in $\mathbb{R}$ is said to converge to $x ∈ \mathbb{R}$, or $x$ is said to be a limit of $(x_n)$, if for every $\epsilon> 0$, there exists a natural number $K ∈ \mathbb{N}$ such that for all $n ≥ K$, the terms $x_n$…
Charlie
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Evaluate $\lim_{x\to\infty}\left[x - \sqrt[n]{(x - a_1)(x - a_2)\ldots(x - a_n)}\right]$

Evaluate the following the limit: $$\lim_{x\to\infty}\left[x - \sqrt[n]{(x - a_1)(x - a_2)\ldots(x - a_n)}\right]$$ I tried expressing the limit in the form $f(x)g(x)\left[\frac{1}{f(x)} - \frac{1}{g(x)}\right]$ but it did not help.
sourish
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