1. Finding Limits Graphically and Numerically
Chapter 2: Limits and Continuity

2. What you’ll learn about
The definition of a limit
Estimating a limit using a numerical or graphical approach
One-Sided and Two-Sided Limits
Different ways that a limit can fail to exist

3. Definition of Limit There are 3 basic approaches to finding limits:
Graphically
Numerically
Algebraically ( or Analytically)
In this section, we will be exploring the first two approaches

4. lim 𝑥→1 𝑥 3 −1 𝑥−1 Example 1) Find
What this problems is asking us is: How do the values of the function behave when x approaches 1, whether or not the function is defined when x = 1?
In this case here, the function is obviously not defined when x = 1 so let’s take a look at the graph of the function to see what is happening near x = 1.
Based on the graph, it appears that the values of the function are both approaching the same value from the left and the right as we get closer to x = 1.
Now let’s use the table function in our calculators to see if we can determine the limit.

5. Ex 1. – continued
Looking at the table, we can see that as we approach x = 1 from both above and below 1, the value of the function approaches 3.
We have explored the limit graphically and numerically (by looking at the table) and it appears that
lim 𝑥→1 𝑥 3 −1 𝑥−1 = 3
A couple of things to note here:
It doesn’t matter that the function is not defined at x = 1. We care about what happens as we get really, really close to x = 1, not at the actual point x = 1.
Please be aware that most of the time your calculator will show a complete graph even when a function is not defined at a certain point. This is an inevitable flaw in the calculator. ALWAYS check the table values to confirm what you see on the graph.

6. Definition of Limit continued
Notice that in the following graphs, each function has a different value at x = 1 yet all 3 functions have the same lim 𝑥→1 because near x = 1 each functions approaches the same value.

7. One-Sided and Two-Sided Limits
NOTE: If a problem does not specify one or two sided, two sided is implied.

8. Find each of the following values.

9. Limits That Fail to Exist
There are 3 main ways that a limit can fail to exist.
𝐥𝐢𝐦 𝒙→𝟎 |𝒙| 𝒙 does not exist (D.N.E. for short) because the behavior of the function differs from the left and the right.
𝐥𝐢𝐦 𝒙→𝟎 𝟏 𝒙 𝟐 does not exist because the graph displays unbounded behavior. (Later we can say that the limit is ∞, but for now we will say D.N.E.)
𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟏 𝒙 does not exist because the function oscillates between two fixed values as x approaches 0.

10. Find each of the following values.

11. Find each of the following values.

12. Summary
If f(x) approaches a limit as x →𝑐 , then the limit value L is unique.
If f(x) does not approach a limit as x →𝑐 , we say lim 𝑥→𝑐 𝑓 𝑥 does not exist.
The limit may exist even if f(c) is not defined.
One sided limits:
lim 𝑥→ 𝑐 − 𝑓 𝑥 =𝐿 if f(x) converges to L as x approaches c through values less than c.
lim 𝑥→ 𝑐 + 𝑓 𝑥 =𝐿 if f(x) converges to L as x approaches c through values greater than c.
The limit exists if and only if both one-sided limits exist and are equal.
*On the AP exam, you MUST show that a limit exists by showing that both one sided limits exist and are equal.
A limit will also fail to exist if
f(x) increases or decreases without bound as x approaches c.
f(x) oscillates between two fixed values as x approaches c.
There is a more formal definition of a limit that you should read on page 60 of your book.