## Limit by epsilon-delta proof: Example 2

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 . » Read more