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
2
votes
3 answers

Upper bound for expression involving logarithms

Let $N = 2^p$ for some $p \in \mathbb{N}$. Find the smallest upper bound for $\frac{N}{2}\log\left(\frac{N}{2}\right) + \frac{N}{4}\log\left(\frac{N}{4}\right) + \ldots + 1$ I guess I could first rewrite this to…
postasaquest
  • 23
2
votes
2 answers

Why am I getting $1$ instead of $e$?

I'm trying to use a calculator to verify the following formula $$ e=\lim_{n\to\infty} (1 + (1/n))^n. $$ According to the calculator the value of $e$ to 9 decimal place accuracy is $2.718281828$. When I plug in $n=1000000000$ above, I get that…
coffeefeet
  • 29
2
votes
1 answer

What is the limit of $\lim\limits_{n\to+\infty}\bigl(n\bigl(1+\frac{1}{n}\bigr)^n-ne\bigr)$ using the squeeze theorem?

What is the limit of $$\lim_{n\to+\infty}\left(n\left(1+\frac{1}{n}\right)^n-ne\right)$$ using the squeeze theorem, without the use of differentiation, integrals and series? I don't know how to get rid of that '$n$' that is multiplying the whole…
Account_For_Education
  • 49
2
votes
2 answers

$\lim\limits_{x\to\infty}\frac{\sin(x)}{\sqrt{x^{2}+1}}$

I have to evaluate the following limit using L'Hospital's rule. $$\lim_{x\to\infty}\frac{\sin(x)}{\sqrt{x^{2}+1}}$$ But when I try to derivate I always get a $\cos(x)$ or $\sin(x)$ function which has no limit when $x\to\infty$. So, how am I supposed…
mvfs314
  • 2,017
  • 15
  • 19
2
votes
1 answer

Finding limit of $\lim_{n\to\infty}\dfrac{(\tan x)^{2n}+x^2}{\sin^2x+(\tan x)^{2n}}$ at $x={\frac{\pi}4}^+$

Let $f:(-\frac{\pi}2,\frac{\pi}2)\to\mathbb R$ $$f(x)=\begin{cases}\lim_{n\to\infty}\dfrac{(\tan x)^{2n}+x^2}{\sin^2x+(\tan x)^{2n}};& x\ne0\\1; & x=0\end{cases}, n\in\mathbb N$$ Which of the following hold(s) good? (A)…
aarbee
  • 8,246
2
votes
4 answers

Limit of $\left(y+\frac23\right)\mathrm{ln}\left(\frac{\sqrt{1+y}+1}{\sqrt{1+y}-1} \right) - 2 \sqrt{1+y}$

We consider the limit for large $y$ of the following expression : $$\left(y+\frac23\right)\mathrm{ln}\left(\dfrac{\sqrt{1+y}+1}{\sqrt{1+y}-1} \right) - 2 \sqrt{1+y}.$$ Many references state that the large $y$ behaviour of this last expression is…
user817256
2
votes
0 answers

Calculate $\lim_{n \to \infty} \frac{(\sqrt{n}+3n)^2-9n^2}{n^{3/2}+7n}$

$$\lim_{n \to \infty} \frac{(\sqrt{n}+3n)^2-9n^2}{n^{3/2}+7n}$$ I'm stuck and would appreciate some assistance :D Here's what I did so far: $$ \begin{split} \lim_{n \to \infty} \frac{(\sqrt{n}+3n)^2-9n^2}{n^\frac{3}{2}+7n} &= \lim_{n \to \infty}…
jenny
  • 87
2
votes
2 answers

Calculating the Limit of a Function that divisor approaches to zero

I try to calculate the limit of a function: $$ f : \mathbb{R} \to \mathbb{R}, f(x) = \begin{cases}x^2-3 , x \geq 0 \\ 3x, x < 0\end{cases}$$ Evaluate $ \lim_{x\to2} \frac{f(x)-f(2)}{x-2}.$ A)$0\ \ \ \ $ B)$4\ \ \ \ $ C)$2\ \ \ \ $ D)$3\ \ \ \ $ E)…
tahasozgen
  • 215
2
votes
1 answer

If a limit of a fraction exists and has a finite value, and the denominator approaches zero, then can we say that the numerator also approaches zero?

Suppose a real valued function $$h(x) = \frac{f(x)}{g(x)}$$. And we know that $$\lim_{x\to a} h(x) = L$$ Where $L$ is real and finite. We also know that $$\lim_{x\to a}g(x) = 0$$ Can we infer from this that $\lim_{x\to a}f(x) = 0$ ? If $f(x)$…
watch
  • 35
  • 6
2
votes
2 answers

Epsilon-N proof with a subtraction in the denominator

I'm in an introductory-level analysis class and am really struggling to grasp epsilon-N proofs, the way our professor has taught them (as opposed to simpler versions I've learned in the past) just makes no sense to me and does not seem to apply…
user924523
  • 21
  • 2
2
votes
1 answer

Find all values of parameter $a$ for the limit to exist.

Let $\left \lfloor {x} \right \rfloor$ to be floor value of $x$. $$ \mathop {\lim }\limits_{x \to 0} \left( {\frac{{\arcsin \left\lfloor {(a + 2)x} \right\rfloor }}{{x + a}} - \cos \left\lfloor {\left| {ax} \right|} \right\rfloor } \right). $$ I…
qweasdzxcqweasd
  • 31
2
votes
1 answer

Proof involving the precise definition of a limit: is this wrong?

I'm having some trouble with a proof involving the precise definition of a limit. I need to do the following: Show that there exists $\delta > 0$ such that: $\left|f(x,y)-f(0,0)\right|<\epsilon$ whenever $\sqrt{ x^2 + y^2 }<\delta$ with…
Dale
  • 21
  • 1
2
votes
2 answers

Manipulating limit expressions when you are told that the limit of one of the expressions does not exit (Spivak - Chapter 5 Problem 23)

Questions 23 a) and b) from Chapter 5 of of Spivak's Calculus are written as follows: a) Suppose that $\displaystyle\lim_{x \to 0}f(x)$ exists and is $\neq 0$. Prove that if $\displaystyle\lim_{x \to 0}g(x)$ does not exist, then…
S.C.
  • 4,984
2
votes
6 answers

How to compute limit similar to the one of $e$.

How do I compute the following limit: $$\lim\limits_{n\to \infty} \left(1+\dfrac{1}{\sqrt{n}}\right)^{\sqrt n} $$ I'm very certain that the above converges to $e$ but don't really know how I can show that rigorously.
Jacob
  • 333
2
votes
2 answers

Limit derivation of generalized mean, $-\infty$ case yields contradiction with ones as input

Consider the following function (a CES utility function or weighted power mean): $$u_\rho(\vec x) = \left(\sum_{i=1}^n a_i x_i^\rho\right)^{1/\rho}$$ for parameters $a_i \geq 0$, $\sum a_i =1$, and $\rho\leq1$. It can be shown (see the Wikipedia…
Max
  • 1,257
1 2 3
…
99 100