1. Unit 4 – Exponential, Logarithmic, and Inverse Trigonometric Functions
Lesson 7
Chapter 4.5: L’H o pital’s Rule and Indeterminate Forms

2. Thought we were done with limits? NEVER!
Today’s lesson deals with the limits of certain types of rational functions, and how to use a new rule to evaluate their limits with the help of derivatives.
Before we start, I thought it would be good to review a few concepts that you will need to know when working with such limits.

3. Special Trig Limit Theorems:

4. Limits at Infinity Theorems
In both cases, the limit for both scenarios is zero!
𝑛 is odd
𝑛 is even
Does your understanding of HAs help?

5. More Limits at Infinity Theorems
𝑛 is odd
𝑛 is even
In both cases, the limit for both scenarios is +∞!

6. More Limits at Infinity Theorems
𝑛 is odd
𝑛 is even
lim 𝑥→−∞ 𝑥 3 =−∞
lim 𝑥→−∞ 𝑥 4 =+∞

7. L’Hôpital’s Rule
Actually, L’Hôpital’s Rule was developed by his teacher Johann Bernoulli. De l’Hôpital paid Bernoulli for private lessons, and then published the first Calculus book based on those lessons.
Guillaume De l'Hôpital
Greg Kelly, Hanford High School, Richland, Washington

8. L’Hôpital’s Rule
The REAL author!
Johann Bernoulli

9. Consider:
If we try to evaluate this by direct substitution, we get:
Zero divided by zero can not be evaluated, and is an example of indeterminate form.
In this case, we can evaluate this limit by factoring and canceling:

10. The limit is the ratio of the numerator over the denominator as x approaches 2.
If we zoom in far enough, the curves will appear as straight lines.

11. As
becomes:

12. As
becomes:

13. L’Hôpital’s Rule:
If is indeterminate, then:

14. Steps YOU need to take in order to use L’H o pital’s Rule successfully!
Check that the limit of 𝑓(𝑥) 𝑔(𝑥) is indeterminate. If not, then L’H o pital’s Rule cannot be used.
Differentiate 𝑓(𝑥) and 𝑔(𝑥) separately.
Find the limit of 𝑓 ′ (𝑥) 𝑔 ′ (𝑥) .
If the limit is a finite # or ±∞, then your answer to the limit of 𝑓(𝑥) 𝑔(𝑥) is the finite # or ±∞ .
If the limit is yet another indeterminate, repeat the process (take the derivative of the derivative).

15. If it’s no longer indeterminate, then STOP!
Good to know
lim 𝑥→𝑎 𝑓 𝑥 𝑔 𝑥 = 0 # =0
lim 𝑥→𝑎 𝑓 𝑥 𝑔 𝑥 = # 0 =±∞
Example:
= 0 1 =0
If it’s no longer indeterminate, then STOP!
If we try to continue with L’Hôpital’s rule:
which is wrong, wrong, wrong!

16. On the other hand, you can apply L’Hôpital’s rule as many times as necessary as long as the fraction is still indeterminate:
(Rewritten in exponential form.)
not

17. Confirm the limits are indeterminate and evaluate using L’Hopital’s rule
lim 𝑥→0 sin 2𝑥 𝑥
= lim 𝑥→0 2 cos 2𝑥 1  =2
lim 𝑥→0 𝑒 𝑥 −1 𝑥 3
= lim 𝑥→0 𝑒 𝑥 3𝑥 2 
= 1 0 =+∞
lim 𝑥→ 𝜋 − sin 𝑥 cos 𝑥
= 0 1 =0
=lim 𝑥→ 𝜋 2 − cos 𝑥 − sin 𝑥 

18. Confirm the limits are indeterminate of the type 𝟎 𝟎 and evaluate using L’H 𝐨 pital’s rule
lim 𝑥→ 0 − tan 𝑥 𝑥 2
= lim 𝑥→ 0 − sec 2 𝑥 2𝑥 
= 1 0 =−∞
lim 𝑥→0 1− cos 𝑥 𝑥 2
= lim 𝑥→0 sin 𝑥 2𝑥 
= lim 𝑥→0 cos 𝑥 2
= 1 2

19. Confirm the limits are indeterminate of the type ∞ ∞ and evaluate using L’H 𝐨 pital’s rule
lim 𝑥→∞ 𝑥 𝑒 𝑥
= lim 𝑥→∞ 1 𝑒 𝑥 =0
Still ∞ ∞
lim 𝑥→ ln 𝑥 csc 𝑥
= lim 𝑥→ 𝑥 − csc 𝑥 cot 𝑥
No problem!
Just rewrite! 
= lim 𝑥→ − 1 𝑥 sin 𝑥 tan 𝑥
= lim 𝑥→ − sin 𝑥 𝑥 tan 𝑥
=− 1 0 =0
lim 𝑥→0 sin 𝑥 𝑥 =1

20. L’Hôpital’s rule can be used to evaluate other indeterminate
forms besides
The following are also considered indeterminate:
The first one, , can be evaluated just like
The others must be changed to fractions first.
The BC class will be shown how to handle the last 5 if time allows

21. 4.5 Homework
Pp /1-14 all

22. Handling the other indeterminate forms
This next part, though not too complicated, is not in the BC curriculum.
However, it is taught in college calculus courses, so why not teach it here?

23. This approaches
This approaches
We already know that
but if we want to use L’Hôpital’s rule:

24. This is indeterminate form
If we find a common denominator and subtract, we get:
Now it is in the form
L’Hôpital’s rule applied once.
Fractions cleared. Still

25. L’Hôpital again.

26. Indeterminate Forms:
Evaluating these forms requires a mathematical trick to change the expression into a fraction.
We can then write the expression as a fraction, which allows us to use L’Hôpital’s rule.
When we take the log of an exponential function, the exponent can be moved out front.
Then move the limit notation outside of the log.
We can take the log of the function as long as we exponentiate at the same time.

27. Indeterminate Forms:
Example:
L’Hôpital
applied p