# How do you find lim_(theta->0) tantheta/theta using l'Hospital's Rule?

May 9, 2017

The derivative of $\tan \theta$ is ${\sec}^{2} \theta$ and the derivative of $\theta$ is $1$.

${\lim}_{\theta \rightarrow 0} \tan \frac{\theta}{\theta} = {\lim}_{\theta \rightarrow 0} {\sec}^{2} \frac{\theta}{1} = {1}^{2} / 1 = 1$

May 9, 2017

Determine that the expression evaluated at the limit results in an indeterminate form.
Differentiate the number and denominator.
The limit of the new expression is the same as the original.

#### Explanation:

The expression evaluated at $\theta = 0$ is the indeterminate form $\frac{0}{0}$, therefore we differentiate the numerator and denominator:

${\lim}_{\theta \rightarrow 0} \frac{\frac{d \left(\tan \left(\theta\right)\right)}{d \theta}}{\frac{d \left(\theta\right)}{d \theta}}$

${\lim}_{\theta \rightarrow 0} {\sec}^{2} \frac{\theta}{1} = {\sec}^{2} \left(0\right) = 1$

Therefore, the limit of the original expression is same:

${\lim}_{\theta \rightarrow 0} \frac{\tan \left(\theta\right)}{\theta} = 1$