All Questions
95
questions
0
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0
answers
31
views
Continuous extension of $f(x,y)=\left(\frac{1+y+x^2}{y}\right)^y$
The function $f(x,y)=\left(\frac{1+y+x^2}{y}\right)^y$ is not defined for $y=0$, so to extend $f$ continuously I want to define $f(x,0)$ as :
$$f(x,0)=\lim _{y \rightarrow 0} \left(\frac{1+y+x^2}{y}\...
- 918
1
vote
1
answer
59
views
Domain of the first derivative
I have a question about a specific exercise. Given the following function
$$f(x) = |x|sin(x^2)$$
we have that the domain is $(-\infty,+\infty)$. Now the first derivative is
$$f'(x) = \frac{xsin(x^2)}{|...
- 33
0
votes
0
answers
73
views
Find discontinuities of the function
My function is $y= \sqrt{(1-\cos(πx))/(4-x^2)}$.
The main question I have problems with answering is wherever 2 and -2 are removable discontinuities and why.
Should I think about the domain of my ...
- 103
-1
votes
1
answer
80
views
Image of $\frac{2x}{\ln(x)}$
I'm trying to calculate the image of $f(x)=\frac{2x}{\ln(x)}$. I tried to find the horizontal asymptotes. But I couldn't. the $\lim_{x \to \infty} f(x)$ is $\infty$. I also tried to find the oblique ...
- 47
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
$$
\lim\limits_{n\to\...
- 783
0
votes
2
answers
76
views
Question about limits and logarithms of functions
Let $f$ be a smooth function of one-variable. Suppose that the limit $\lim_{T\rightarrow \infty}\frac{1}{T}\log f(T)$ exists. Now I am trying to see wether or not we will have that if we do a time ...
- 4,767
0
votes
0
answers
69
views
Prove that a function is constant. [duplicate]
It is an open problem (still open) published on a Chinese mathematics magazine, but i am asking this just because i can not figure it out. I wonder if someone can help? I won't use any answers here to ...
- 83
0
votes
0
answers
18
views
periodic functions, proof that lim x → + ∞ sin(x) does not exist [duplicate]
I'm trying to figure out how to proof the next thing:
Let f be a periodic function in real numbers,that is there exists T> 0 such that f (x) = f (x + T) ∀ x ∈ R. Show that if lim x → + ∞ f (x) ...
1
vote
1
answer
433
views
Show that $f(a+h)=f(a) + hf'(a+t_{h}h)$ and $\lim_{h \to 0} t_{h} = 1/2$
Let $f$ be a function of $C^{2}$ in an interval of the form $[a-h,a+h]$ where $h>0$.
Show that if $f''(a)\ne0$ then there exists $t_{h} \in [-1,1]$ such that $$f(a+h)=f(a) + hf'(a+t_{h}h) \quad, \...
- 431
0
votes
1
answer
128
views
Alternative proof of $x^x \geq \sin x$ if $x>0$
My lecturer gave me this exercise: "Show that for $x>0$ it is $x^x \geq \sin x$."
This is my approach: since $x \geq \sin x$ for all $x \geq 0$, it is sufficient to show that $x^x \geq x$;...
- 2,162
1
vote
1
answer
76
views
$\lim_{x\to\infty} (\sqrt{x+1} + \sqrt{x-1}-\sqrt{2x})$ using little o function
Problem has to be solved specifically using little o function. I was going to transform $\sqrt{x+1}$ into $1+\frac{1}{2}x+o(x)$ and $\sqrt{2x}$ into $1+\frac{1}{2}t+o(t)=1+\frac{1}{2}(2x-1)+o(2x)$ but ...
- 87
0
votes
2
answers
77
views
Find $\lim_\limits{x\to 0}\frac{1-(\cos x)^{\sin x}}{x}$ using little o
$\lim_\limits{x\to 0}\frac{1-e^{\sin(x) \ln(\cos x)}}{x}=\lim_\limits{x\to 0}\frac{1-e^{(x+o(x))\ln(1-\frac{x^2}{2}+o(x^2))}}{x}=\lim_\limits{x\to0}\frac{1-e^{(x+o(x))(-\frac{x^2}{2}+o(x^2))}}{x}$. If ...
- 87
-2
votes
1
answer
37
views
I need help with this simple problem in little-o notation [duplicate]
$\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
-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
3
votes
2
answers
93
views
Knowing the limit of $f'(x)$ find the limit of $f(x)$
We have that $f$ is differentiable on $(a, +\infty)$ with $a>0$.
I want to show that if $\displaystyle{\lim_{x\rightarrow +\infty}f'(x)=\ell}$, then there are the following cases:
If $\ell>0$...
- 14k