# Difficult Question in Calculus — limits and integrals

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1. Mar 25, 2015

### omeraz100

1. The problem statement, all variables and given/known data

(hebrew) : f(x) a continuous function. proof the following

2. Relevant equations
I guess rules of limits and integrals

3. The attempt at a solution
I've tried several approaches:
taking ln() of both sides and using L'Hospitale Rule.
Thought about using integral reduction formula.
But really nothing even got me close.

2. Mar 25, 2015

### Staff: Mentor

Please show us at least some of what you've tried. Per forum rules, you must show an attempt.

3. Mar 25, 2015

### omeraz100

4. Mar 25, 2015

### Dick

Last edited by a moderator: May 7, 2017
5. Mar 25, 2015

### Ray Vickson

If I were doing it would let
$$M = \max_{x \in [a,b]} |f(x)|, \; L_t = \left( \int_a^b |f(x)|^t \, dx \right)^{1/t},$$
then I would show that: (1) $\lim_{t \to \infty} L_t \leq M$; and (2) $\lim_{t \to \infty} L_t \geq M$.

(1) is quite easy; (2) is a bit trickier and needs some properties of continuous functions on finite, closed intervals.