12.1 - Introduction to Limits

Limits are fundamental to calculus, and this week we have started learning about them. To begin, here is the basic definition of a limit from our textbook:

I am going to use an example from our textbook to demonstrate how to find a limit using a table. The problem is asking us to evaluate the following limit:

First, notice that x is approaching 2. Therefore, we will plug in numbers very close to 2 in order to find the value of the limit. We can try numbers such as 1.99, 1.999, 2.01, 2.001, and so on.

3(1.99) – 2 = 3.97

3(1.999) – 2 = 3.997

3(2) – 2 = ?

3(2.01) – 2 = 4.03

3(2.001) – 2 = 4.003

From these calculations, it appears as though f(x) gets closer to 4 as x gets closer to 2. So, we can conclude that the limit is 4.

We could have also solved this specific problem by substituting the x in the equation with 2, the value which x is approaching:

3(2) – 2 = 6 – 2 = 4

Some limits also fail to exist. Here are the conditions under which a limit will not exist:

Finally, here are the properties of limits and operations that can be performed with limits:

All of these pictures can be found in section 12.1 of our textbook.