All Questions
95
questions
28
votes
3
answers
2k
views
What kind of "mathematical object" are limits?
When learning mathematics I tend to try to reduce all the concepts I come across to some matter of interaction between sets and functions (or if necessary the more general Relation) on them. Possibly ...
- 1,191
13
votes
1
answer
4k
views
Lim Sup/Inf for real valued functions
To understand the notion of, say, limit superior for a sequence, is not difficult. Simply consider the set of all upper buonds for the set of all limit points of the sequence, and then simply pick the ...
- 760
11
votes
5
answers
1k
views
(Non)Existence of limits
When we say that a limit of a function does not exist in $\mathbb{R}$ (or some metric space) does it make sense to say that it might exist somewhere else?
[I am trying to think along lines of ...
- 584
11
votes
3
answers
368
views
Regularity of the function $|x|^ax$
Assuming $x \in \mathbb{R}$, what can we say about the regularity class ($C, C^1, C^2, ..., \text{or}\ C^\infty$) of the following function (also with respect to $a \in \mathbb{R}$)? $$f(x)=|x|^ax$$
- 1,764
8
votes
1
answer
339
views
Prove that $\lim _{n \rightarrow \infty} \sqrt{n} \cdot\sin^{\circ n}(\frac{1}{\sqrt{n}})=\frac{\sqrt{3}}{2}$ [duplicate]
It's known that $\lim _{n \rightarrow \infty} \sqrt{n} \cdot \sin^{\circ n}(x)=\sqrt{3}$ for any $x>0$.
And I found a new conclusion $\lim _{n \rightarrow \infty} \sqrt{n} \cdot \sin^{\circ n}(\...
- 133
6
votes
5
answers
258
views
Calculating limit $\lim\limits_{x\to\infty}\frac{3x^2-\frac{3}{x^2+1}-4f'(x)}{f(x)}$ for an unknown function.
Given that $f(x)$ is a continuous function and satisfies $f'(x)>0$ on $(-\infty,\infty)$ and $f''(x)=2 \forall x \in(0,\infty)$.We need to find the limit
$$\lim_{x\to\infty}\frac{3x^2-\frac{3}{x^...
- 241
6
votes
5
answers
17k
views
If $f(x)<g(x)$ prove that $\lim f(x)<\lim g(x)$
I have this question:
Let $f(x)→A$ and $g(x)→B$ as $x→x_0$. Prove that if $f(x) < g(x)$ for all $x∈(x_0−η, x_0+η)$ (for some $η > 0$) then $A\leq B$. In this case is it always true that $A &...
- 403
6
votes
1
answer
239
views
If $a_n=\sqrt{1+\sqrt{2+\cdots\sqrt{n}}}$ and $\lim\limits_{n\to\infty}a_n=\ell$, prove $\lim_{n\to\infty}[(\ell-a_n)^{1/n}n^{1/2}]=\frac{\sqrt e}2$
$$a_n=\sqrt{1+\sqrt{2+\cdots\sqrt{n}}}$$
We can prove that $\{a_n\}$ is convergent
(using mathematical induction, $\sqrt {k+\sqrt{k+1+\cdots\sqrt{n}}}\leq k-1, for \ k\geq3$).
If
$$
\lim\limits_{n\to\...
- 783
4
votes
1
answer
158
views
Limit of sequence $\lim_{n\to\infty}\frac{1+(\sqrt{n}+1)^{3}+2\sqrt{n}}{n+\sin(n)}$
This is no homework. It's another task of a sample exam and I'd like to know how to solve it.
Find the limit of $$\lim_{n\to
\infty}\frac{1+(\sqrt{n}+1)^{3}+2\sqrt{n}}{n+\sin(n)}$$
Both numerator ...
- 4,710
4
votes
4
answers
5k
views
Prove that a polynomial diverges to infinity.
I would like to prove the following statement:
Let $P$ be a polynomial of degree $n$ where $n$ is an odd natural number and $x$ $\in$ $\mathbb{R}$. $P(x)=a_{0}+a_{1}x+ ... + a_{n}x^{n}$
If $a_{n} &...
- 119
4
votes
0
answers
127
views
Under what conditions is $\lim_{x\to a}\left|\varphi\circ f(x)-\tau \circ g(x)\right|=0$ true?
This question is inspired from another much easier problem I was trying to solve which I tried to generalize. The question is essentially as follows (assuming all the limits exist)
If $a\in \mathbb R\...
- 9,592
3
votes
2
answers
1k
views
Composition of limits
Find functions $f,g : \mathbb R \longrightarrow \mathbb R $ and $a,b,c \in \mathbb R $ such that
$$ \lim_{x \rightarrow a} f(x) = b \quad \text{and} \quad \lim_{y \rightarrow b} g(y) = c \quad \text{...
- 33
3
votes
1
answer
75
views
Give $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $|f'(x)|<1$ $ f(x) \neq x$ for all $x \in \mathbb{R}$
Problem
Give a function $f:\mathbb{R} \rightarrow \mathbb{R}$ , $C^\infty$ such that
$1) |f'(x)|<1$
$2) f(x) \neq x$ for all $x \in \mathbb{R}$
My ideia
The idea is to get a function that ...
- 931
3
votes
1
answer
43
views
Help with the study of the function $f(x) = \frac{-2}{5x-\ln\vert x \vert}$
I'm having problems in understanding few things about this function, also because some of my calculations do not match the plot.
$$f(x) = \dfrac{-2}{5x-\ln\vert x \vert}$$
Here is what I did.
First of ...
- 3,482
3
votes
4
answers
522
views
Calculate the limit: $\lim_{x\rightarrow \infty}\frac{\ln x}{x^{a}}$
Calculate the limit: $$\lim_{x\rightarrow \infty}\frac{\ln x}{x^{a}}$$
When try calculate limit, we get $\frac{\infty}{\infty}$, so use L'Hôpital again.
$$(\ln x)' = \frac{1}{x}$$
$$x^{a} = e^{\ln ...
- 1,007