L’Hospital’s rule and indeterminate quotients

Understand the idea

Before applying L’Hospital’s rule, verify that the quotient approaches $\displaystyle 0/0$ or an infinite-over-infinite form, and that the differentiability and denominator-derivative conditions hold on the relevant punctured interval. Replace the quotient by the quotient of derivatives when the theorem applies. Other forms need algebraic rewriting first. Recheck the form before applying the rule again.

\lim\frac{f(x)}{g(x)}=\lim\frac{f'(x)}{g'(x)}\quad\text{under L’Hospital conditions}

Worked example

Evaluate $\displaystyle \lim_{x\to0}(e^{2x}-1)/x$ .

The numerator and denominator both approach zero. Near zero the functions are differentiable and the denominator derivative is

\displaystyle 1

. L’Hospital gives

\displaystyle \lim_{x\to0}2e^{2x}=2

.

Try it yourself

Evaluate $\displaystyle \lim_{x\to\infty}\ln x/x$ .

Show solution and reasoning

The original quotient is an infinite-over-infinite form. The derivative quotient is $\displaystyle (1/x)/1$ , which tends to zero.