All Questions
9
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
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
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
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
$$ \lim_{x \rightarrow a} f(x) = b \quad \text{and} \quad \lim_{y \rightarrow b} g(y) = c \quad \text{...
- 33
0
votes
1
answer
159
views
$f:[-1,1]\to \mathbb{R}$ is a continuous function such that $f(2x^2-1)=(x^3+x)f(x).$ Find $\lim_{x\to 0}\frac {f(\cos x)}{\sin x}$
$f:[-1,1]\to \mathbb{R}$ is a continuous function such that $f(2x^2-1)=(x^3+x)f(x).$ Find $\lim_{x\to 0}\frac {f(\cos x)}{\sin x}$
My approach : Using the functional equation I got f is an odd ...
- 2,269
-3
votes
1
answer
53
views
Could anyone explain to me how did we get this result? This is simple example but I'm struggling with little o notation in general. [closed]
$\left(x-\displaystyle\frac{x^3}{6}+\displaystyle\frac{x^5}{120} +o(x^5)\right)\left(1+\displaystyle\frac{x^2}{2}-\displaystyle\frac{x^4}{24}+ \displaystyle\frac{x^4}{8}+o(x^4)\right) = x+\...
- 87
11
votes
3
answers
368
views
Regularity of the function $|x|^ax$
Assuming $x \in \mathbb{R}$, what can we say about the regularity class ($C, C^1, C^2, ..., \text{or}\ C^\infty$) of the following function (also with respect to $a \in \mathbb{R}$)? $$f(x)=|x|^ax$$
- 1,764
4
votes
4
answers
5k
views
Prove that a polynomial diverges to infinity.
I would like to prove the following statement:
Let $P$ be a polynomial of degree $n$ where $n$ is an odd natural number and $x$ $\in$ $\mathbb{R}$. $P(x)=a_{0}+a_{1}x+ ... + a_{n}x^{n}$
If $a_{n} &...
- 119
-2
votes
2
answers
112
views
How do I prove interval 𝐴⊂[0,3] exists on this integration
Let $f:[0,3] \to \mathbb{R}$ be a continuous function satisfying
$$\int_{0}^{3}x^{k}f(x)dx=0 \quad \text{for each k = 0,1,}\dots,n-1$$ and
$$\int_{0}^{3}x^{n}f(x)dx=3.$$
Then prove that there is an ...
- 45