# Thread: show the limit is b

1. ## show the limit is b

show that if $0 b$

2. Originally Posted by wopashui
show that if $0 b$
$b = \sqrt[n]{{b^n }} \leqslant \sqrt[n]{{b^n + a^n }} \leqslant \sqrt[n]{{b^n + b^n }} = b\sqrt[n]{2}$

3. Originally Posted by Plato
$b = \sqrt[n]{{b^n }} \leqslant \sqrt[n]{{b^n + a^n }} \leqslant \sqrt[n]{{b^n + b^n }} = b\sqrt[n]{2}$
this just proved it's bounded, is it?

4. Originally Posted by wopashui
this just proved it's bounded, is it?
What is $\lim_{n\to\infty} b$ ?

What is $\lim_{n\to\infty} b \sqrt[n]{2}$ ?

What does that tell you then if $b \le \sqrt[n]{a^n + b^n} \le b\sqrt[n]{2}$ ?

5. Originally Posted by wopashui
this just proved it's bounded, is it?
Apply the squeeze theorem.