1. Limits of Radical and Trig Functions
Lesson 1.1.9

2. With any limit, you can always graph and/or make a table of values.
However, there are more exact and less tedious shortcuts, as we saw yesterday.
Today, we will learn shortcuts to be used for rational and trigonometric functions.

3. Learning Objectives
Given a function with a radical binomial, multiply the numerator and denominator by the conjugate of that binomial to evaluate the limit at a certain point.
Given a trigonometric function, evaluate the limit at a certain point.

4. Indeterminate Forms of Limits
Suppose that we were asked to evaluate the limit on the right.
What would happen if we plugged in 0 for x?

5. We would end up with the fraction 0/0.
Of course, such a fraction is undefined.
When plugging in c gives us 0/0 or ±∞/ ±∞, we say that the limit is in indeterminate form.

6. When a limit is in this form, we can determine it by manipulating the function in some way.
One of those ways is by rationalizing.

7. Rationalizing
In Algebra II, you learned to rationalize fractions with radicals to eliminate a radical in the denominator.
In Calculus, you will instead rationalize to evaluate a limit.

8. You can rationalize either the numerator or the denominator.
You no longer have to worry about not leaving radicals in the denominator

9. Review: What is Rationalizing?
Rationalizing comes from the difference of squares concept. (a+b)(a-b) = a2 – b2
Notice how a+b and a-b are the same thing, but with the middle sign changed. They are called conjugates of each other.
Keep in mind: when you square a square root, the radical sign goes away:

10. Therefore
To rationalize, multiply numerator and denominator by the conjugate.

11. Rationalizing Practice

12. Example 1 Find the limit on the right. First rationalize.
Then plug in.

13. Trigonometric Limits
Please know the two limits on the right. (Don’t worry about why.)

14. Example 2
Evaluate the following limit:

15. Trig Identities
Other trig limits may require you to apply trigonometric identities. Know the ones below.

16. Reteaching #1
Evaluate the following limit

17. Example 3

18. Graphs of Trig Functions
For some trig limits, it helps to refer to the graph of one of the basic trig functions.
We went over sine and cosine in Lesson
Now let’s go over tan, cot, sec, csc.

19. y = tan x
Note: the vertical lines are asymptotes. They are not part of the graph.

20. y = cot x

21. y = sec x

22. y = csc x

23. Example 4 Use trig identities to simplify.
Use one of your graphs to determine.

24. Homework
Textbook 1a,c; 2a; 3a,c,d