All Questions
95
questions
0
votes
1
answer
88
views
How do I solve this limit without L'Hospital? [duplicate]
$$\lim_{x\to0}\frac{6^x-1}{x}$$
I have spent quite a lot of time on this limit but I still can't solve it. None of the regular tricks work here, I can't get rid of the $x$, nor can I get it in the ...
- 1,798
2
votes
5
answers
209
views
How do I solve this limit without l'Hopital?
I tried the substitution $t=x-(\pi/3)$ but it doesn't help at all. I have also tried using $\sin(\pi/3)=\sqrt{3}/2$ but couldn't do anything useful then. I tried to factor the denominator and ...
- 1,798
1
vote
1
answer
98
views
Continuous function and a related infimum
Let $f : [a, b) \rightarrow \mathbb{R}$ be continuous with $f(a) = 0$ and assume that there is some $x' \in ( a, b)$ such that $f(x') > 0$. Denote by
$$
x_0 := \inf \{ x \in [ a, b) : f ( x ) > ...
- 1,557
0
votes
1
answer
40
views
limit of a function defintion
Definition(limit of a function):
$f:D \subset \mathbb{R}^n \to \mathbb{R}^m$, $x_0 \in \mathbb{R}^n$ limit point of $D$
Then $ \lim_{x \to x_0}f(x)=y$ means that for each sequence $x_n \in D\...
user727154
0
votes
1
answer
66
views
Let f:(0,∞) -> R be a function such that f(1)=0, f is differentiable at 1 and f'(1)=1. Suppose that f(xy)=f(x)+f(y) for all x,y in (0,∞)
prove that lim(h->0) (f(1+h))/h = 1.
prove that f is differentiable on (0,∞) and find a formula for f'.
prove that (f(e^x)-x)'= 0 for all x in R.
I am unsure where to start with all of these ...
- 99
1
vote
2
answers
40
views
Can I determine the limit of these functions by inserting?
Given:
$$\lim_{x \rightarrow \infty}\left(\frac{ex+3}{x+e}\right)^{(x^2-1)/(2x^2+6x+4)},$$
$$\lim_{x \rightarrow 1-}(1-x)^{1-x^2}.$$
My ideas would be:
it holds that $\lim_{x \rightarrow \infty} \...
- 635
1
vote
2
answers
54
views
$ \lim_{n \to \infty} \int_0^2 f_n(x)dx$ and $ \lim_{n \to \infty} f_n(x) \text{ for } x \in [0,2]$
For $n \in \mathbb{N}$ let $f_n:[0,2] \to \mathbb{R}$ be defined by
$$f_n(x) =
\left\{
\begin{array}{ll}
n^3x^2, \text{ if } 0 \leq x \leq \frac{1}{n} \\
2n - n^2x \text{ if } \frac{1}{n} < x \...
- 931
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
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
1
vote
3
answers
60
views
Why if $\lim_{x \to a}f(x)=b$ then $\lim_{n \to \infty}f(x_n)=b$? [closed]
Why is that if $$\lim_{x \to a}f(x)=b \Rightarrow \lim_{n \to \infty}f(x_n)=b?$$
This is often used to prove that the same properties that apply to sequences, also apply to functions, but I don't ...
- 1,412
1
vote
2
answers
51
views
find limit of a multivariable function
I have to show continuity at $(0,0)$ of $f(x,y)=\frac{\sin(x^2) + \sin(y^2)}{\sqrt{x^2 +y^2}}$ for $(x,y)\ne(0,0)$ and $f(0,0)=0$.
I tried to find the limit using polar coordinates
$ \frac{\sin(r^2\...
- 41
0
votes
1
answer
69
views
Does $\lim_{x\to 0} \frac {f(x)}{|x|} \leq M $ ,for some real $M$ ,imply that $f(x)\leq |x| M $?
Just to restate my question as in the title:
Does $\lim_{x\to 0} \frac {f(x)}{|x|} \leq M $ , for some real $M$, imply that $f(x)\leq |x| M $ ?
Any help would be appreciated!
user489116
-2
votes
2
answers
45
views
$\lim\limits_{x \to 0} \frac{\sqrt{(a^4+(bx)^2} - \sqrt{a^4+b}}{x}$
calculate...
this is my homework
the photo is my solution proposal but I don't know what to do next....
- 175
0
votes
2
answers
64
views
Proof: All directional derivatives $\frac{\partial f}{\partial e}$ of $\frac{sin(x^3+y^3)}{x^2+y^2}$ are in the origin
Let $M := (0,\infty) \subset \mathbb{R^2}$ and $f:\mathbb{R}^2 \to \mathbb{R}$.
How can one prove that all directional derivatives $\frac{\partial f}{\partial e}$ of $f(x,y)$ are existing in the ...
user616397
1
vote
0
answers
26
views
Finding functions using cosine
Assume that the sine and cosine functions are continuous at the point 0.
(a) Find functions $a(x), b(h), c(x), d(h)$ such that $b(h)$ and $d(h)$ are continuous at $h = 0$ and $b(0) = d(0) = 0$, such ...
- 139