If the derivative tends to $0$ at infinity, then so does the function divided by $x$
44060
If $\lim_{x \to +\infty} f'(x) = L$, then $\lim_{x \to \infty} \frac {f(x)}{x} = L$
121217
How to show that $\lim\limits_{x \to \infty} f'(x) = 0$ implies $\lim\limits_{x \to \infty} \frac{f(x)}{x} = 0$?
23615
Why does zero derivative imply a function is locally constant?
11167
A counterexample of a differentiable function on an interval whose derivative is discontinuous
68156
For a differentiable function $f:\mathbb{R} \rightarrow \mathbb{R}$ what does $\lim_{x\rightarrow +\infty} f'(x)=1$ imply? (TIFR GS $2014$)
85207
Must the derivative of the wave function at infinity be zero?
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.
68752
A few questions about infinite derivative.
10428
Proving that $\lim\limits_{x\to\infty}f'(x) = 0$ when $\lim\limits_{x\to\infty}f(x)$ and $\lim\limits_{x\to\infty}f'(x)$ exist
113343
$f$ is constant if derivative equals zero
12556
A non-trivial, non-negative, function bounded below by its derivative with $f(0)=0$?
11385
Using the Limit definition to find the derivative of $e^x$
813
$f$ uniformly continuous and $\int_a^\infty f(x)\,dx$ converges imply $\lim_{x \to \infty} f(x) = 0$
131444
Local maximum implies derivative is $0$
129579
If $f''(x_0)$ exists then $\lim_{x \to x_0} \frac{f(x_0+h)-2f(x_0)+f(x_0-h)}{h^2} = f''(x_0)$
139216
Use the definition of derivative to find $f'(x)$ for $f(x) = x^{1/2}$.
89180
limit of reciprocal of function