All Questions
95
questions
2
votes
3
answers
78
views
Find the limit of $x(x + 1 - \sin(\frac{1}{1+x})^{-1})$ as $x \rightarrow \infty$
As the title states, I need to find the limit for $x\left(x + 1 - \frac{1}{\sin(\frac{1}{1+x})}\right)$ as $x \rightarrow \infty$, as part of a larger proof I am working on.
I believe the answer is 0....
- 467
1
vote
1
answer
85
views
Does the derivative of $f(x)=\begin{cases}\dfrac{x}{2}+x^2\sin\left(\dfrac{1}{x}\right)& x\neq 0,\\0, &x=0\end{cases}$ exist everywhere?
I'm trying to prove that
$$f(x)=\begin{cases}\dfrac{x}{2}+x^2\sin\left(\dfrac{1}{x}\right)& x\neq 0,\\0, &x=0.\end{cases}$$
has a derivative everywhere. Here is what I have done:
Let $x_0\...
- 4,504
0
votes
1
answer
107
views
Product of convex functions with special properties
Let $f(x)$ and $g(x)$ be non-negative, convex functions in $C^2([M,\infty))$, where $M > 0$. Also, assume $f(x)$ is strictly decreasing on $[M,\infty)$, and that $g(x)$ is strictly increasing on $[...
- 69
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
$$\lim_{x\to\infty}\frac{3x^2-\frac{3}{x^...
- 241
1
vote
1
answer
151
views
Showing that Derivative is Linear
Question is: In $$f(a+h) - f(a) = h f'(a + \frac h 2), \qquad a, h \in \mathbb R$$ show $f'$ is line.
I have no problems with the first part. I'm however having trouble with taking the derivative ...
- 1,522
0
votes
4
answers
373
views
Analysis: Show that there exists an $x$ so that $f'(x)=0$. (Derivative at point is equal to$ 0$) [duplicate]
Question: Look at already answered question. Same.
- 1,522
0
votes
2
answers
707
views
How show that function is greater or equal zero? [closed]
Hi maths peoples I have question how you show that function is greater or equal to zero because I want show that function is dense function and this is one of two condition for show it is dense ...
- 1,335
1
vote
0
answers
279
views
Generalize Squeeze Theorem
Theorem. Let $I$ be an interval having the point $a$ as a limit point. Let $g$, $f_1$, $f_2$ ,..., $f_n$ and $h$ be functions defined on $I$, except possibly at $a$ itself. Given that for every $x$ in ...
- 5,872
0
votes
1
answer
30
views
Limit of maximum of a function on an interval [closed]
Let $f:[0,\infty)\to\mathbb{R}$ be a bounded function such that $f(x) \to 0$, as $x \to \infty$. Prove that $$\max\limits_{u \in [x/2,x]}f(u) \to 0$$ as $x \to \infty$.
I need some help.
- 4,761
-1
votes
1
answer
49
views
$ f:[0,1] \to \mathbb{R} , f(x)={1\over2x+1}$ [closed]
$$ f:[0,1] \to \mathbb{R} , f(x)={1\over2x+1}$$
If we apply Lagrange on $[0,x]$ , $x$ $\in (0,1)$, we obtain the point $c(x) \in (0,x)$.
If $$l=\lim_{x\to 0} {c(x) \over x}$$ Then $$l= ?$$
Some help ...
- 195
4
votes
4
answers
5k
views
Prove that a polynomial diverges to infinity.
I would like to prove the following statement:
Let $P$ be a polynomial of degree $n$ where $n$ is an odd natural number and $x$ $\in$ $\mathbb{R}$. $P(x)=a_{0}+a_{1}x+ ... + a_{n}x^{n}$
If $a_{n} &...
- 119
0
votes
4
answers
3k
views
Prove that $\lim_{x\to\infty} (\ln x) = \infty$
Can someone help me prove that the function $\ln(x)$ diverges to infinity as $x$ approaches infinity. I tried using the definition to show that $\lvert \ln(x) -∞ \rvert < \epsilon $ where $\epsilon ...
- 99
0
votes
1
answer
31
views
identifying discontinuity
In the following function, how do you qualitatively show that the function is discontinuous?
$$f(x) = \frac{\sqrt{1 - \sqrt{\sin(2x)}}}{\pi - 4x}$$
I started off by analyzing the numerator and ...
- 299
0
votes
1
answer
26
views
Can't prove the integral of two variables converges
Let $f:(0,1]\to\mathbb R$ be continuous and positive, and:
$$\lim_{t\to 0}f(t) = +\infty,\quad \int_0^1 f(t)dt = \lim_{\epsilon\to 0^+}\int_\epsilon^1 f(t)dt<\infty$$
Show that $F:B(0,1)\to\...
- 79
0
votes
2
answers
35
views
For which $a,b$ given function is continous?
For which $a,b\in\mathbb{R}$, is $f(x)$ continuous?
$f:(-1,+\infty)$, $f(x)=\lim\limits_{n\to\infty} \frac{x^2+bx}{a+x^n}$
pre calculus, can't figure out how to analyze continuity at $x=1$
- 5