Draw a unit circle with the vertical tangent, showing clearly the lengths and for some angle in the first quadrant. Can you see that the area of the circular sector is a little greater than the area of the smaller triangle, and a little less than the area of the larger triangle. But as , the areas will all also . That means
Now as and . Since is sandwiched between two quantities that go to 1, it must also go to 1.
Note: Technically we have only proven the right hand limit, where from the positive direction, but the proof is almost identical for the left hand limit, you just need to work in the fourth quadrant and be careful with negatives and the direction the inequality signs point.