$\displaystyle \lim_{x \rightarrow \infty}(x-lnx)$

I made it into indeterminate form so that I can use l'Hospital's rule:

$\displaystyle \lim_{x \rightarrow \infty}(lne^x-lnx) = \lim_{x \rightarrow \infty}ln\left(\frac{e^x}{x}\right)$

I take the derivative and get:

$\displaystyle \frac{x-1}{x}$

Taking the derivative again I get that the function approaches 1, but based off of the actual graph the function approaches infinity. What am I doing wrong?