Evaluating a limit of infinity/infinity form using L'Hopital's Rule

As $x \to \infty$, both the numerator $x + \ln(x)$ and denominator $x - \ln(x)$ tend to $\infty$, so the limit has the indeterminate form $\infty/\infty$. This confirms L'Hopital's Rule can be applied.

Differentiating numerator and denominator separately: the derivative of $x + \ln(x)$ is $1 + \frac{1}{x}$, and the derivative of $x - \ln(x)$ is $1 - \frac{1}{x}$. This gives the new limit $\lim_{x\to\infty} \frac{1 + 1/x}{1 - 1/x}$.

As $x \to \infty$, $1/x \to 0$, so the expression approaches $\frac{1+0}{1-0} = 1$. Therefore the original limit equals $1$.