If f is define as $\displaystyle f(x)=\displaystyle \lim_{n \to \infty} \frac{1}{n} \log(x^{n}+e^{n})$ then is f continuous.
Thanks for the reply. I'm just trying to wrap my mind around the domain of the function. You have an $\displaystyle \infty/\infty$ situation there. l'Hospital's Rule applied once gives you
$\displaystyle \displaystyle{f(x)=\lim_{n\to\infty}\dfrac{e^{n}+x ^{n}\ln(x)}{e^{n}+x^{n}}}.$
If you divide everything through by $\displaystyle e^{n}$, you could simplify as follows:
$\displaystyle \displaystyle{f(x)=\lim_{n\to\infty}\dfrac{1+(\fra c{x}{e})^{n}\ln(x)}{1+(\frac{x}{e})^{n}}}.$
Now, the domain we can see to be $\displaystyle (0,\infty).$
The behavior of this function will change depending on whether $\displaystyle x\le e$ or not. For $\displaystyle 0<x<e,$ the function is going to return 1. For $\displaystyle x=e,$ the function will return 1. For $\displaystyle x>e,$ I think you could apply l'Hospital's rule again.
That's one approach.
I think no need to apply Hospital's a second time.
If we look at the second form provided by Ackbeet:
if $\displaystyle x<e$, we get 1 which is continuous
if $\displaystyle x=e$, we get $\displaystyle \frac{1+ln(x)}{2}$ also continuous.
if $\displaystyle x>e$, we get: $\displaystyle \displaystyle{f(x)=\lim_{n\to\infty}\dfrac{1+(\fra c{x}{e})^{n}\ln(x)}{1+(\frac{x}{e})^{n}} = \lim_{n\to\infty}\dfrac{(\frac{x}{e})^{n}\ln(x)}{( \frac{x}{e})^{n}} = ln(x)} $. which is clearly continuous.
hope this helps