Very special limits to know!
The following two famous limits can be used to find many other limits:
$lim_{x \to 0} \frac{sin x}{x} = 1$ and $lim_{x \to 0} \frac{cos x - 1}{x} = 0$ .

The next theorem gives an additional property of limits.

The Squeeze Theorem
If $f (x) \leq h (x) \leq g (x)$ when x is near c (except possibly at c) and
$lim_{x \to c} f (x) = lim_{x \to c} g (x) = L$ ,
then $lim_{x \to c} h (x) = L$ .

We only need to know that $f (x) \leq h (x) \leq g (x)$ is true around $x = c$ , because we are working with limits. They are only concerned with what is going on around $x = c$ , not what is actually happening at $x = c$ .

Now, if we again assume that all three functions are nice enough — again, this isn't required to make the Squeeze Theorem true; it only helps with the visualization — then we can get a quick sketch of what the Squeeze Theorem is telling us. The following figure illustrates what is happening in this theorem.

From the figure we can see that if the limits of $f (x)$ and $g (x)$ are equal to $L$ at $x = c$ then the function values must also be equal at $x = c$ . This is where we're using the fact that we assumed the functions where “nice enough”, which isn't really required for the Theorem. However, because $h (x)$ is “squeezed” between $f (x)$ and $g (x)$ at this point then h(x) must have the same value. Therefore, the limit of $h (x)$ at this point must also be the same.

The Squeeze theorem is also known as the Sandwich Theorem or the Pinching Theorem.