This is the overdelayed continuation of the discussion on the definition of limits. In this post, we discuss another example.

Prove that the .

Recall that the definition states that the limit of as approaches if for all , however small, there exists a such that if , then .

From the example 1, we have learned that we should manipulate , to make one of the expressions look like . Solving, we have

.

Note that we have accomplished our goal, going back to the definition, this means that if , then .

Now, it is not possible to divide both sides by (making it because varies. This means that we have to find a constant such that .

If is confined to some interval centered at , then we can find . For instance, suppose , which is the same as , then . In particular .

Therefore,

. Hence, we have . But we have two restrictions:

and

so, to be sure that both inequalities are obeyed, we choose to be smaller than and abbreviated as whenever .