What's wrong with the following limit evaluation?

Question

\begin{align*} \lim_{x \to \infty} \ln{x}&=\lim_{x \to \infty}\frac{\ln{x}}{x}\cdot x\\ &=\lim_{x \to \infty}\frac{\ln{x}}{x}\cdot \lim_{x \to \infty} x\\ &=\lim_{x \to \infty}\frac{(\ln{x})'}{x'}\cdot \lim_{x \to \infty} x\text{ [Using l'hopital]}\\ &=\lim_{x \to \infty} 1/x \cdot \lim_{x \to \infty} x=\lim_{x \to \infty} 1/x\cdot x=1 \end{align*}