Does (a_n)^{b_n} converge to a^b as n approaches infinity?

[no_alone]

Homework Statement

a_n -> a
b_n -> b
prove that a_n^b_n = a^b
?
Its all Sequence of course

 

[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".

 

[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 $$n \rightarrow \infty$$