# An^bn as an->a and bn->b

1. Nov 11, 2009

### no_alone

1. The problem statement, all variables and given/known data
an -> a
bn -> b
prove that anbn = ab
?
Its all Sequence of course

2. Nov 11, 2009

### JG89

Hint: $$(a_n)^{b_n} = e^{b_n log(a_n)}$$

Now use continuity of the exponential and logarithm function to take the limits "inside the function".

3. Nov 11, 2009

### JG89

By the way, we don't want to prove that $$(a_n)^{b_n} = a^b$$. We want to prove that $$(a_n)^{b_n} \rightarrow a^b$$ as $$n \rightarrow \infty$$