If $\lim_{x \rightarrow \infty} f(x) + f'(x) = 0$, prove $\lim_{x\to \infty} f(x)$ and $\lim_{x \rightarrow \infty} f'(x)$ exist and both equal $0$.
67364
If $\lim_{x\to\infty}(f(x)+f'(x))=L$ show that $\lim_{x\to\infty} f(x) = L$ and $\lim_{x\to\infty} f'(x) = 0$
58361
Is there a continuous function such that $\int_0^{\infty} f(x)dx$ converges, yet $\lim_{x\rightarrow \infty}f(x) \ne 0$?
111205
Proving that $\lim_{k\to \infty}\frac{f(x_k)-f(y_k)}{x_k-y_k}=f'(c)$ where $\lim_{k\to \infty}x_k=\lim_{k\to \infty}y_k=c$ with $x_k<c<y_k$
70458
Show that it is possible that the limit $\displaystyle{\lim_{x \rightarrow +\infty} f'(x)} $ does not exist.
63889
Prove that if $u_n\rightarrow \infty$ then $\frac{1}{u_n} \rightarrow 0$
44060
If $\lim_{x \to +\infty} f'(x) = L$, then $\lim_{x \to \infty} \frac {f(x)}{x} = L$
6990
For $f$ Riemann integrable prove $\lim_{n\to\infty} \int_0^1x^nf(x)dx=0.$
144422
How to prove that $\lim\limits_{x\to\infty} f(x)/x=L$
20675
$L=\lim_{x\to\infty}(f(x)+f'(x))$ exists . Which of the following statements is\are correct?
44336
Is $\lim\limits_{x\to x_0}f'(x)=f'(x_0)$?
57600
Show $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0 $
133405
$\lim_{x \rightarrow 0} f'(x) = L$ exists, does it follow that $f'(0)$ exists?
21879
Show that if $\lim_{x\to a}f'(x) = A$ then $f'(a)$ exists and equals A
71127
How do I find $f(0)$, $f'(0)$, and $f'(x)$ given $f(x+y)=f(x)+f(y)+x^2y+xy^2$ and $\lim_{x\to0}\frac{f(x)}{x}=1$?
83707
Suppose that $f$ is differentiable on $\mathbb{R}$ and $\lim_{x\to \infty}f'(x)=M$. Show that $\lim_{x\to \infty}f(x+1)-f(x)$ exists and find it.
34187
The limit of the derivative of an increasing and bounded function is always $0$?
44370
If $f(x)\to 0$ as $x\to\infty$ and $f''$ is bounded, show that $f'(x)\to0$ as $x\to\infty$