ThE LIMIt OF A FUNCTION THE SANDWICH THEOREM In the previous section, we examined eleven limit laws and have illustrated each law with several examples. Note that the limit laws are also the properties of limits. Before we conclude this tutorial, we take a look at two additional properties of limits (or limit laws), expressed as Theorems.
If f(x) ≤ g(x) when x is close to, but NOT equal to a and the limits of f and g both exist, then,
lim f(x)
x→a
≤
lim g(x)
x→a
Theorem 1 The next law is based on Theorem 1, and is called the Sandwich Theorem. This Theorem is stated thus:
If f(x) ≤ g(x) ≤ h(x) when x is close to, but NOT equal to a and
then
lim f(x)
=
lim g(x)
=
x→a
x→a
lim h(x)
x→a
=
L
L Theorem 2
You're probably wondering why this Theorem is called a “Sandwich” Theorem. Perhaps this graph would partly explain why: