Answer
$$\lim_{x\to\infty}(x-\ln x)=\infty$$
Work Step by Step
$$A=\lim_{x\to\infty}(x-\ln x)$$
We need to do some modifications here, which would change $x$ into a logarithmic function somehow.
We see that $\ln e^x=x$. So we can rewrite $x$ with $\ln e^x$.
$$A=\lim_{x\to\infty}(\ln e^x-\ln x)$$
$$A=\lim_{x\to\infty}\ln\Big(\frac{e^x}{x}\Big)$$
$$A=\ln\Big(\lim_{x\to\infty}\frac{e^x}{x}\Big)$$
$$A=\ln B$$
We see that as $x\to\infty$, $e^x\to\infty$. So we have an indeterminate form of $\infty/\infty$, applicable to L'Hospital's Rule:
$$B=\lim_{x\to\infty}\frac{(e^x)'}{x'}$$
$$B=\lim_{x\to\infty}\frac{e^x}{1}$$
$$B=\lim_{x\to\infty}e^x$$
$$B=\infty$$
Therefore, $$A=\ln B=\ln(\infty)=\infty$$
