L'Hopital's Rule Math Example 1

Follow the full solution, then compare it with the other examples linked below.

easy

Find

\displaystyle\lim_{x \to 0} \frac{\sin x}{x}

using L'Hôpital's rule.

1
Direct substitution: $\frac{\sin 0}{0} = \frac{0}{0}$ — indeterminate form.
2
Apply L'Hôpital: differentiate numerator and denominator separately.
3
$\lim_{x\to 0}\frac{\sin x}{x} = \lim_{x\to 0}\frac{\cos x}{1} = \frac{\cos 0}{1} = 1$ .

Answer

1

L'Hôpital's rule applies when we have a

\frac{0}{0}

form. Differentiate top and bottom independently (not using the quotient rule), then evaluate the limit.

About L'Hopital's Rule

\lim_{x \to a} \frac{f(x)}{g(x)}

is an indeterminate form

\frac{0}{0}

\frac{\infty}{\infty}

, then

\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}

provided the right-hand limit exists (or is

\pm\infty

Find [formula].

Find [formula] using L'Hôpital's rule.

Find [formula].