Limit with the sine
A standard limit with the sine reads
Explanation
We use the unit circle
and draw three surfaces in it. We're going to determine the limit by comparing those surfaces.
The surface of a triangle is , and of a sector
, so
| triangle OCA | = | ||
| sector ODA | = | ||
| triangle OEA | = |
In the drawing you can clearly see the relation of the surfaces, and that is
Multiplying with 2 gives
and dividing by sin θ gives
If θ → 0+ than cos θ → 1 and from that follows
so that
This is an important trigonometric limit.