Question

The limit as x approaches infinity of (e^x+x)^(1/x). I got that it diverges, but I'm not sure if I made a mistake.
My work:
lim(e+x^(1/x))
lim(e+(1/x^x))
lim(ex^x+1)/x^x
l'hopital:lim(e^(e(lnx+1))+1)/e^(lnx+1)
diverges?

Answer 1

I would do this as follows. Take the logarithm of the function:

1/x Log[x + exp(x)] =

1/x {Log(exp(x)) + Log[1 + x exp(-x)]} =

1/x {x + Log[1 + x exp(-x)]}

Using that log(1+epsilon) =
expsilon+ O(epsilon^2), we can write this as:

1 + exp(-x) + O(xexp(-2x))

So, the limit of the logarithm is of the function is 1, therefore the limit is e (this is true because the logarithm is a continuous function).