Answer
$$m=\frac{\sin x}{x}$$
The slope of the secant line joining $(0,0)$ to the following point:$$x=0.1 \quad \rightarrow \quad m \approx 0.99833417 \\ x=0.01 \quad \rightarrow \quad m \approx 0.99998333$$
The exact slope of the tangent line at the point $(0,0)$ is $m=1$.
Work Step by Step
The slope of the secant line joining the points $(x, \sin x)$ and $(0,0)$ can be obtained as follows.$$m=\frac {\Delta y }{ \Delta x} \quad \Rightarrow \quad m=\frac {y_2-y_1}{x_2-x_1}= \frac{\sin x -0}{x-0}=\frac{\sin x}{x}.$$Evaluating this formula at $x=0.1$ and $x=0.01$, we get$$x=0.1 \quad \rightarrow \quad m= \frac{\sin 0.1}{0.1} \approx 0.99833417 \\ x=0.01 \quad \rightarrow \quad m= \frac{\sin 0.01}{0.01} \approx 0.99998333 \, .$$To find the exact slope of the tangent line at the point $(0,0)$, we must find the limit of $m$ when $x$ approaches $0$; that is,$$\lim_{x \to 0}\frac{\sin x}{x}=1 .$$
