$\lim_{x \rightarrow 0} f'(x) = L$ exists, does it follow that $f'(0)$ exists?
1371
Prove that $f'(a)=\lim_{x\rightarrow a}f'(x)$.
101731
What is $\lim_{(x,y)\rightarrow(0,0)} \frac{x^3-y^3}{x^2-y^2}$?
64381
Prove that if one of $\lim_{x \rightarrow 0 }{f(x)}$ and $\lim_{x \rightarrow 0}{f(x^3)}$ exists, then the other one also exists.
107795
If $\,f''(x) \ge f(x)$, for all $x\in[0,\infty),$ and $\,f(0)=f'(0)=1$, then is $\,f(x)>0$?
67292
Prove that $\lim_{t\to \infty} t\mu(\{x:f(x)\geq t\})=0$
130161
Showing that if $\lim_{x\to\infty}xf(x)=l,$ then $\lim_{x\to\infty}f(x)=0.$
136738
Does the limit $\lim _{x\to 0} \frac 1x \int_0^x \left|\cos \frac 1t \right| dt$ exists?
127983
Does $\lim_{x \to 0+} \left(x\lfloor \frac{a}{x} \rfloor\right)=a?$
155188
How do I prove this limit does not exist: $\lim_{x\rightarrow 0} \frac{e^{1/x} - 1}{e^{1/x} + 1} $
119602
$f:\mathbb{R} \to \mathbb{R}$ be differentiable and $\lim\limits_{x\to\infty}f'(x)=1$, is $f(x)$ unbounded?
47381
If $\lim_{x\to\infty} [f(x+1)-f(x)] =l$ then $\lim_{ x\to\infty}f(x)/x =l$ ($f$ is continuous)
60498
Prove that $\lim_{(x,y)\rightarrow(0,0)} \frac{|x|^{a}|y|^{b}}{|x|^{c} + |y|^{d}}$ does not exist
30204
Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$
33337
$\int_{0}^{\infty} f(x) \,dx$ exists. Then $\lim_{x\rightarrow \infty} f(x) $ must exist and is $0$. A rigorous proof?
145233
How to show $\lim_{n \rightarrow \infty} \int_0^1 n x^n f(x) \; dx = f(1)$ for continuous $f$?
158152
Given $f: \Bbb R\rightarrow\Bbb R$ and a point $a\in\Bbb R$. Prove $lim_{x\rightarrow a} f(x)=\lim_{h\rightarrow 0} f(a+h)$ if 1 of the limits exists.
86409
Show that $\lim_{x \to +\infty}\left(f(x)+f'(x)\right)=0 \Rightarrow \lim_{x \to +\infty} f(x)=0$