How to prove that both f(x) and its derivative decay to zero, as x grows to infinity,
6620
If $\lim\limits_{x\rightarrow\infty} (f'(x)+f(x)) =L<\infty$, does $\lim\limits_{x\rightarrow\infty} f(x) $ exist?
28297
derivative of x^x^x... to infinity?
35325
Prove that $\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0$.
71225
$f(0)=0$ and $\lim_{x \to \infty}f(x)=0$. Show there's $x_0$ such that $f'(x_0)=0$
42817
proving $\lim\limits_{n\to\infty} \int_{0}^{1} f(x^n)dx = f(0)$ when f is continous on [0,1]
24471
Does the derivative of a continuous function goes to zero if the function converges to it?
48464
If the derivative approaches zero then the limit exists
115712
Application of Derivatives rigorous proof
145573
How to show that $f$ is a zero function?
57193
Prove that $f$ has derivatives of all orders at $x=0$, and that $f^{(n)}(0)=0$ for $n=1,2,3\ldots$
117906
Show that $\liminf\limits_{x\to \infty}|f'(x)|=0$ if $\lim\limits_{x\to \infty}f(x)/x=0$ and $f$ is differentiable
105700
Proving $\,f$ is constant.
105068
Proving that the function is differentiable and $f'(0)=0$
85207
Must the derivative of the wave function at infinity be zero?
12556
A non-trivial, non-negative, function bounded below by its derivative with $f(0)=0$?
130161
Showing that if $\lim_{x\to\infty}xf(x)=l,$ then $\lim_{x\to\infty}f(x)=0.$
138544
For which values of $x$, will $f (x) . f '(x) < 0$?