13. Compute the limit by using the definition of the derivative. $$ \lim _{x ightarrow 0} \frac{ an x}{x}[\operatorname{Hints}: an 0=0 ext{ and } x=x-0 ].$$

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13. Compute the limit by using the definition of the derivative. $$ \lim _{x ightarrow 0} \frac{	an x}{x}[\operatorname{Hints}: 	an 0=0 	ext{ and } x=x-0 ].$$

UpStudy AI · Accepted Answer

We rewrite the limit as
$$
\lim_{x \rightarrow 0} \frac{\tan x - \tan 0}{x - 0}.
$$
Since $$\tan 0 = 0$$, this is the definition of the derivative of $$\tan x$$ at $$x=0$$. That is,
$$
\lim_{x \rightarrow 0} \frac{\tan x}{x} = \tan'(0).
$$

The derivative of $$\tan x$$ is given by
$$
\frac{d}{dx} \tan x = \sec^2 x.
$$
Evaluating at $$x=0$$,
$$
\tan'(0) = \sec^2 0.
$$
Since $$\sec x = \frac{1}{\cos x}$$ and $$\cos 0=1$$, we have
$$
\sec^2 0 = \left(\frac{1}{1}\right)^2 = 1.
$$

Thus, the limit is
$$
\lim_{x \rightarrow 0} \frac{\tan x}{x} = 1.
$$
The limit is 1.

Compute the limit by using the definition of the derivative.

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