There are several limits of special interest involving trigonometric functions.
1.
This limit can be proven with the squeeze theorem.
For
Dividing everything by
which reduces to:
Taking the limit of the right-hand side:
The squeeze theorem tells us that:
Equivalently:

Sinc Function
The normalized sinc (blue, higher frequency) and unnormalized sinc function (red, lower frequency) shown on the same scale.
2.
This equation can be proven with the first limit and the trigonometric identity
We start with:
Multiplying the numerator and denominator by
Using the algebraic limit theorem,
Therefore:
3.
This relation can be proven by substituting