1. Lecture 5: L’Hôpital’s Rule, Sequences, and Series

2. Part I: L’Hôpital’s Rule

3. Indeterminate forms
How do we compute lim 𝑥→𝑎 𝑓(𝑥) 𝑔(𝑥) when 𝑓 𝑎 =𝑔 𝑎 =0?
Similarly, how do we compute lim 𝑥→∞ 𝑓(𝑥) 𝑔(𝑥) when lim 𝑥→∞ 𝑓 𝑥 = ±∞ and lim 𝑥→∞ 𝑔 𝑥 = ±∞ ?

4. Computing Indeterminate forms
If 𝑓 𝑎 =𝑔 𝑎 =0, 𝑓 ′ 𝑎 and 𝑔′(𝑎) exist, and 𝑔′(𝑎)≠0 then lim 𝑥→𝑎 𝑓(𝑥) 𝑔(𝑥) = 𝑓 ′ (𝑎) 𝑔 ′ (𝑎)
To see this, note that
lim 𝑥→𝑎 𝑓(𝑥) 𝑔(𝑥) = lim ∆𝑥→0 𝑓 𝑎+∆𝑥 −𝑓(𝑎) 𝑔 𝑎+∆𝑥 −𝑔(𝑎) = lim ∆𝑥→0 𝑓 𝑎+∆𝑥 −𝑓(𝑎) ∆𝑥 ∙ lim ∆𝑥→0 ∆𝑥 𝑔 𝑎+∆𝑥 −𝑔(𝑎) = 𝑓 ′ (𝑎) 𝑔 ′ (𝑎)
What happens if 𝑓 ′ 𝑎 = 𝑔 ′ 𝑎 =0?

5. L’Hǒpital’s Rule L’Hǒpital’s Rule:
If 𝑓 𝑎 =𝑔 𝑎 =0 then lim 𝑥→𝑎 𝑓(𝑥) 𝑔(𝑥) = lim 𝑥→𝑎 𝑓 ′ (𝑥) 𝑔 ′ (𝑥) provided that the limit on the right exists and 𝑔′(𝑥)≠0 when 𝑥 is close to 𝑎 but not equal to it.
L’Hǒpital’s Rule for limits at infinity:
If lim 𝑥→∞ 𝑓 𝑥 = ±∞ and lim 𝑥→∞ 𝑔 𝑥 = ±∞ then lim 𝑥→∞ 𝑓(𝑥) 𝑔(𝑥) = lim 𝑥→∞ 𝑓 ′ (𝑥) 𝑔 ′ (𝑥) provided that the limit on the right exists and 𝑔′(𝑥)≠0 when 𝑥 is sufficiently large. A similar statement holds for limits at −∞.

6. L’Hǒpital’s Rule Examples
lim 𝑥→∞ 2𝑥 2 +3𝑥+4 𝑥 2 = lim 𝑥→∞ 4𝑥+3 2𝑥 = lim 𝑥→∞ 4 2 =2 lim 𝑥→0 sin 𝑥 −𝑥 𝑥 3 = lim 𝑥→0 cos 𝑥 −1 3𝑥 2 = lim 𝑥→0 −sin 𝑥 6𝑥 = lim 𝑥→0 −cos 𝑥 6 = −1 6

7. Part II: Sequences and Series

8. Objectives Know how to find the limit of sequences
Know how to compute geometric series and telescoping series
Corresponding sections in Simmons: 13.2, 13.3

9. Sequences
If we have an element 𝑥 𝑛 for every n, this is called a sequence
Example: 1,2,3,4,… 𝑥 𝑛 =𝑛
Example: 1, 1 2 , 1 3 , 1 4 ,… 𝑥 𝑛 = 1 𝑛

10. Convergence of Sequences
We say a sequence 𝑥 1 , 𝑥 2 ,… converges to L if lim 𝑛→∞ 𝑥 𝑛 =𝐿
Limits of sequences can be found in the same way as limits of functions. L’Hopital’s rule often works, sometimes other tools are needed.
Example: If 𝑥 𝑛 = 𝑛 2 +2𝑛 2𝑛 , lim 𝑛→∞ 𝑥 𝑛 = 1 2
Example: If 𝑥 𝑛 = 𝑛 2 +2𝑛 −𝑛, complete the square. 𝑥 𝑛 = 𝑛 2 +2𝑛 −𝑛= 𝑛+1 2 −1 −𝑛 𝑥 𝑛 ≈𝑛+1− 1 2 𝑛+1 −𝑛 so lim 𝑛→∞ 𝑥 𝑛 =1

11. Series
A series is an expression of the form 𝑛=1 ∞ 𝑥 𝑛 = 𝑥 1 + 𝑥 2 + 𝑥 3 + 𝑥 4 +…
Can be viewed as a sequence 𝑦 𝑛 = 𝑗=1 𝑛 𝑥 𝑗
Key example: Geometric series
Geometric series have the form
1+𝑎+ 𝑎 2 + 𝑎 3 +… (times a constant)

12. Evaluating Series Geometric series can be evaluated as follows
1+𝑎+…+ 𝑎 𝑛−1 = 1−𝑎 𝑛 1−𝑎
𝑛=1 ∞ 𝑎 𝑛−1 = 1 1−𝑎 if and only if 𝑎 <1
Partial fractions and cancelling terms can also be useful
𝑛=1 ∞ 1 𝑛 2 +𝑛 = 𝑛=1 ∞ 1 𝑛 − 1 𝑛+1 = 𝑛=1 ∞ 1 𝑛 − 𝑛=2 ∞ 1 𝑛 =1
When terms cancel like this, it is called a telescoping series.