1. Finding Limits Graphically and Numerically

2. To sketch graph, we need to know what’s going on at 𝑥=1.
Ex 𝑓 𝑥 = 𝑥 2 −3𝑥+2 𝑥−1
To sketch graph, we need to know what’s going on at 𝑥=1.
We can use a table:
𝑥 approaches 1 from the left.
𝑥 approaches 1 from the right. 𝑥
0.75
0.9
0.99
0.999 1
1.001
1.01
1.1
1.25
𝑓(𝑥)
𝑓(𝑥) approaches −1.
𝑓(𝑥) approaches −1.

3. From the table, we see that 𝑓 𝑥 approaches −1 as 𝑥 gets closer to 1.
We can verify that by looking at the graph of 𝑓(𝑥).

4. This leads us to the definition of the limit.
Def: We write
lim 𝑥→𝑎 𝑓 𝑥 =𝐿
and say “the limit of 𝑓(𝑥) as 𝑥 approaches 𝑎 is 𝐿” if we can make the values of 𝑓 𝑥 as close to 𝐿 as we want by taking 𝑥 to be close to 𝑎 (but not equal to 𝑎) on both sides of a.

5. Three Cases to Consider:
Remark: lim 𝑥→𝑎 𝑓 𝑥 =𝐿 in all three cases.

6. The limit is reinforced by the graph of 𝑓.
Ex 2. Create a table of values for the function and use the result to estimate
lim 𝑥→0 𝑥 𝑥+1 −1 𝑥
𝑓(𝑥)
The limit is reinforced by the graph of 𝑓.

7. Ex 3. Find the limit of 𝑓(𝑥) as 𝑥 approaches 2, where
𝑓 𝑥 = 1, 𝑥≠2 0, 𝑥=2
Remark: The existence or non-existence of 𝑓(𝑥) at 𝑥=𝑎 has no bearing on the existence of the limit of 𝑓(𝑥) as 𝑥 approaches 𝑐.

8. LIMITS THAT FAIL TO EXIST
Ex 4. Show that the limit does not exist.
Remark: The limit fails to exist due to different left and right behavior.

9. lim 𝑥→ 𝑥 2
Ex 5. Discuss the existence of the limit
Remark: The limit fails to exist due to unbounded behavior.

10. lim 𝑥→0 sin 1 𝑥
Ex 6. Discuss the existence of the limit
Remark: The limit fails to exist due to oscillating behavior.

11. So, in summary (from Ex. 4-6), limits fail to exist in three situations:
Different left and right behavior
Unbounded behavior
Oscillating behavior