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

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.

Special Trig Limit Theorems:

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?

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

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

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 1661 - 1704 Greg Kelly, Hanford High School, Richland, Washington

L’Hôpital’s Rule The REAL author! Johann Bernoulli 1667 - 1748

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:

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.

As becomes:

As becomes:

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

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).

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!

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

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 2 1− sin 𝑥 cos 𝑥 = 0 1 =0 =lim 𝑥→ 𝜋 2 − cos 𝑥 − sin 𝑥 

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

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→ 0 + ln 𝑥 csc 𝑥 = lim 𝑥→ 0 + 1 𝑥 − csc 𝑥 cot 𝑥 No problem! Just rewrite!  = lim 𝑥→ 0 + − 1 𝑥 sin 𝑥 tan 𝑥 = lim 𝑥→ 0 + − sin 𝑥 𝑥 tan 𝑥 =− 1 0 =0 lim 𝑥→0 sin 𝑥 𝑥 =1

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

4.5 Homework Pp. 283-284/1-14 all

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?

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

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

L’Hôpital again.

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.

Indeterminate Forms: Example: L’Hôpital applied p