Ben Borgers

Proof that limit as x → 0 of sin(x)/x = 1

grade-12 calculus

This is a unit circle, so there two radii are labeled with length 11. The vertical line labeled sin⁡x\sin{x} because sine of the angle xx is that line over the hypotenuse (11). The second vertical line is labeled tan⁡x\tan{x} because tangent of the angle xx is that line over the base of the triangle (11).

Compare the areas of three regions on the diagram above:

We can see visually that these areas can be arranged in order of size like this:

By the Squeeze Theorem, we can see that sin⁡xx\frac{\sin{x}}{x} is “squeezed” between 1 and cos⁡x\cos{x}. As xx tends to 0, cos⁡x\cos{x} tends to 1 as well.

Therefore, sin⁡xx\frac{\sin{x}}{x} is squeezed between 1 and 1 as x → 0, so the limit of sin⁡xx\frac{\sin{x}}{x} is 1.