If $f\in L^{p_0}[0,1]$, $p_0$ is a given number big enough. Prove: $\lim_{p \to 0+}\Vert f\Vert_p=e^{\int_0^1 \ln\vert f(x)\vert dx}.$
8503
Limit of $L^p$ norm when $p\to0$
38761
Is the $ L^{p}$$[0,1]$ norm continuous in p?
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$?
70790
Evaluation of $\lim_{q \to 0} (\int_0^1 |f|^q)^{1/q} $ for $f \in L^1([0,1])$.
5229
Prove that $\lim_{n \to \infty} \int_0^1{nx^nf(x)}dx$ is equal to $f(1)$.
74170
Prove that $\lim_{x \to +\infty} \frac{f(x)}{x} = L$ if $\lim_{x \to +\infty} [f(x+1) - f(x)] = L \space$
59162
Prove $\liminf\limits_{n\to\infty} F(x_n)\le F(x)$
37015
Prove $\int_X |f|^p=p\int^{\infty}_{0} t^{p-1}\mu({x: |f(x)>t}) dt\,$
121771
Proof $f(x)\equiv 0$
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.
142637
If $f \in \mathcal{L}^1(\mathbb{R})$ then $\lim_{h\rightarrow 0}\int_{-\infty}^\infty |f(x+h)-f(x)|dx\rightarrow 0$
47026
if $f^{(n)}(x_0),g^{(n)}(x_0)$ exist and $\lim_{x\to x_0} \dfrac{f(x)-g(x)}{(x-x_0)^n}=0$ then $f^{(r)}(x_0)=g^{(r)}(x_0 )$ for all $0\leq r\leq n.$
129579
If $f''(x_0)$ exists then $\lim_{x \to x_0} \frac{f(x_0+h)-2f(x_0)+f(x_0-h)}{h^2} = f''(x_0)$
57203
Prove that $\lim_{x \to \infty}\big(\frac{x}{x-1}\big)^x$ is also $e$.
36358
How prove that $\lim\limits_{x\to+\infty}f(x)=\lim\limits_{x\to+\infty}f'(x)=0$ if $\lim\limits_{x\to+\infty}([f'(x)]^2+f^3(x))=0$?
29287
Prove that: $\lim_{n\to\infty} f(n+1) - f(n) = \lim_{x\to\infty} (f(x))' $
26126
$f \in L^1$, but $f \not\in L^p$ for all $p > 1$