L’Hôpital’s rule practice question 20

What do the base $1 + \tfrac{3}{x}$ and the exponent $x$ each tend to? A variable exponent is easier to handle once a logarithm brings it down.

The form is $1^{\infty}$. Write the power as $e^{x\log(1 + 3/x)}$ and find the limit of the exponent, an $\infty\cdot 0$ form.

Rewrite as $\dfrac{\log(1 + 3/x)}{1/x}$, which is $\tfrac{0}{0}$, and apply the rule.