Page 5 - Calc for the Clueless

The limit from the right,

also equals 6. Since the limit from the left equals the limit from the right, the limit exists and is equal to 6. We
write

After seeing this example, you might tell me, "Hey, you big dummy!! All you have to do is substitute x = 3 and
get the answer!!" Substitution does work sometimes and should always be tried first. However, if limits (and
calculus) were so easy, it would not have taken such dynamite mathematicians as Newton and Leibniz to
discover calculus.

Example 2—

We first substitute x = 4 and get 0/0, which is indeterminate. We again make a chart.

x

4.1 1

4.01 1

4.001 1

3.9 1

3.99 1

3.9999 1

As we get close to 4 from both sides, the answer not only is close to I but equals 1. We conclude that the limit
as x goes to 4 equals 1.

We get a little better idea of

This means that f(x) is defined at all points very close to a and that the closer x gets to a, the closer f(x) gets to
L (if it doesn't already equal L).

Example 3—

Nothing bad here.