Suppose that $\underset{x\rightarrow0}{\lim}\frac{f\left(x\right)}{\left|x\right|}=1$ , may $f$ be differentiable at $0$?
136714
Proving that $\lim_{x \to 0} \frac{f(x)}{|x|}=1$ implies $f$ is not differentiable at $0$.
87743
Is a function whose derivative vanishes at rationals constant?
96112
Differentiating $ \left( 1 - \frac {1}{x} \right)^x $
68156
For a differentiable function $f:\mathbb{R} \rightarrow \mathbb{R}$ what does $\lim_{x\rightarrow +\infty} f'(x)=1$ imply? (TIFR GS $2014$)
67791
Can I exchange limit and differentiation for a sequence of smooth functions?
61190
If all derivatives of $f$ are uniformly bounded by a common constant and $f(1/n) = 0$ for all $n$, is $f$ identically zero?
24449
How discontinuous can a derivative be?
12556
A non-trivial, non-negative, function bounded below by its derivative with $f(0)=0$?
129579
If $f''(x_0)$ exists then $\lim_{x \to x_0} \frac{f(x_0+h)-2f(x_0)+f(x_0-h)}{h^2} = f''(x_0)$
137527
if $f(x)$ is differentiable at a x, prove that: $\lim_{h\to0}\frac{f(x+h)-f(x-h)}{2h}$
145913
Find $\lim_{x\rightarrow 0} \frac{\cos x - 1}{x}$
111470
Proof of differentiability of $f(x)$
119907
Finding the Derivative of |x| using the Limit Definition
85614
Do inequalities hold under differentiation?
130748
Infinitely differentiable function with given zero set?
132747
How can one show that $ f(0)\ln(\frac{b}{a})=\lim_{\epsilon\rightarrow 0}\int_{\epsilon a}^{\epsilon b} \frac{f(x)}{x}dx$?
13533
Proof that the limit of $\frac{1}{x}$ as $x$ approaches $0$ does not exist