Answer
The slope of the tangent line at the point $(0,0)$ equals$$m=1 .$$
Work Step by Step
To find the slope of the tangent line at the point $(0,0)$, we should first find the slope of the secant line joining $(x, \tan x)$ and $(0,0)$ as follows.$$m=\frac{\Delta y}{\Delta x} \quad \Rightarrow \quad m=\frac{y_2-y_1}{x_2-x_1}=\frac{\tan x-0}{x-0}=\frac{\tan x}{x} .$$Now, to find the slope of the tangent line at the point $(0,0)$, we must find the limit of $m$ when $x$ approaches $0$; that is,$$\lim_{x \to 0} \frac{\tan x}{x}=\lim_{x \to 0} \left (\frac{\frac{\sin x}{\cos x}}{x} \right )=\lim_{x \to 0} \left (\frac{1}{\cos x} \right )\left (\frac{\sin x}{x} \right )=(1)(1)=1$$(Please note that we have used Theorem 1.9 (1) in finding the limit).
