# What is the limit as x approaches 0 of tanx/x?

Jun 12, 2018

1

#### Explanation:

${\lim}_{x \to 0} \tan \frac{x}{x}$

graph{(tanx)/x [-20.27, 20.28, -10.14, 10.13]}

From the graph, you can see that as $x \to 0$, $\tan \frac{x}{x}$ approaches 1

Jun 12, 2018

Remember the famous limit:

${\lim}_{x \to 0} \sin \frac{x}{x} = 1$

Now, let's look at our problem and manipulate it a bit:

${\lim}_{x \to 0} \tan \frac{x}{x}$

$= {\lim}_{x \to 0} \frac{\sin x \text{/} \cos x}{x}$

$= {\lim}_{x \to 0} \frac{\left(\sin \frac{x}{x}\right)}{\cos x}$

$= {\lim}_{x \to 0} \left(\sin \frac{x}{x}\right) \cdot \left(\frac{1}{\cos} x\right)$

Remember that the limit of a product is the product of the limits, if both limits are defined.

$= \left({\lim}_{x \to 0} \sin \frac{x}{x}\right) \cdot \left({\lim}_{x \to 0} \frac{1}{\cos} x\right)$

$= 1 \cdot \frac{1}{\cos} 0$

$= 1$