sin(x)/x, proof

It is easy to find the limit

\[\lim_{x \rightarrow 0}\frac{\sin x}{x} \]

numerically. If you want to prove what the limit is, you must use geometry.

In order for the limit to become an easy number, you must use radians for measuring angles, this is the reason why degrees are never used when doing calculus. This limit is used for finding the derivative of the trigonometric functions.

Exercise

Using the notations in the applet above:

Find the areas of the triangles \(\Delta OAP\), \(\Delta OBC\), and the area of the sector \( OBP \). Describe the areas in terms of \(\alpha \), \(\sin (\alpha)\) and \(\cos (\alpha)\).

Use the inequalities \(\Delta OAP \lt OBP \lt \Delta OBC \) to find the limit

\[\lim_{\alpha \rightarrow 0}\frac{\alpha}{\sin{\alpha}} \]

Rearrange the inequalities to find the limit

\[\lim_{\alpha \rightarrow 0}\frac{\sin \alpha }{\alpha} \]