Evaluating a limit involves parameters

Question

Let $a,b>0$:
$$\mathop {\lim }\limits_{n \to \infty } {({a^n} + {b^n})^{\frac{1}{n}}}$$

At first look, it seemed simple, yet I couldn't evaluate it.
Maybe Squeeze Thm?

Squeeze theorem works fine here. Suppose without loss of generality $a \geqslant b$. — Daniel Fischer, Jan 25 '14 at 14:33
@B.S., From the second link:
$${\left( {\frac{A}{2}} \right)^x} = \frac{{\log (A)}}{{\frac{1}{x}}}$$ Why is it true? — SuperStamp, Jan 25 '14 at 14:42
@SuperStamp: Great Andrea used the Hopital's rule for solving it. In fact set $y=(\frac{A}2)^x$ and then take $\log$ from both sides and... — Mikasa, Jan 25 '14 at 14:49

score 5 · Accepted Answer · answered Jan 25 '14 at 14:39

5

HINT

Take the largest out of the parentheses (say $a>b$). You then have $a(1+x^n)^{\frac{1}{n}}$ where $x<1$. At this point, you get the limit easily.

I am sure you can take from here.

answered Jan 25 '14 at 14:39

Claude Leibovici

the answer is $a$? Hence $max(a,b)$. Right? – SuperStamp Jan 25 '14 at 14:58
1

@SuperStamp. As far as I know, you are right. – Claude Leibovici Jan 25 '14 at 15:06

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