This is a unit circle, so there two radii are labeled with length . The vertical line labeled because sine of the angle is that line over the hypotenuse (). The second vertical line is labeled because tangent of the angle is that line over the base of the triangle ().
Compare the areas of three regions on the diagram above:
- Blue + yellow:
- The area of that sector of the circle is that part of the circle ( radians out of total radians around the circle) times the area of the circle (, or , so ).
- Blue + yellow + red:
We can see visually that these areas can be arranged in order of size like this:
- Divide each term by
- Simplify the last step
- Take the reciprocal of each term, and therefore flip inequality signs
By the Squeeze Theorem, we can see that is “squeezed” between 1 and . As tends to 0, tends to 1 as well.
Therefore, is squeezed between 1 and 1 as x → 0, so the limit of is 1.