When is it OK to pass the limit to the exponent

[clueless...]

when is it justified to do something like this

$$\lim_{n\to\infty} e^{lnx^{1/n}} = e^{\displaystyle\lim_{n\to\infty}lnx^{1/n}}$$

or something like this

$$\lim_{n\to\infty} 2^{f(n)} = 2^{\displaystyle\lim_{n\to\infty}f(n)}}$$ ?

I am assuming that I can do something like this in both cases, but why?

thank you.

 

[lurflurf]

in general
lim f(g(n))=f(lim g(n))
holds when f is continuous at lim g(n)
exponential functions are everywhere continuous so this can be done in both cases

 

[HallsofIvy]

In fact $\lim_{x\to a}f(x)= f(\lim_{x\to a} x)= f(a)$, from which $\lim_{x\to a}f(g(x))= f(\lim_{x\to a}g(x))$
is the definition of "f is continuous at x= a".

 

[Caesar_Rahil]

Wherever it is continuous which is everywhere