Finding Limits Graphically and Numerically

To sketch graph, we need to know what’s going on at 𝑥=1. Ex 1. 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.

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).

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.

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

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.

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.

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.

lim 𝑥→0 1 𝑥 2 Ex 5. Discuss the existence of the limit Remark: The limit fails to exist due to unbounded behavior.

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

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