Limits of Radical and Trig Functions

Lesson 1.1.9

• 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.

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.

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?

• 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.

• 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.

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.

2

3

x

• You can rationalize either the numerator or the denominator.

• You no longer have to worry about not leaving radicals in the denominator

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: 3)3( 2

Therefore

• To rationalize, multiply numerator and denominator by the conjugate.

2

3

x

2

2

2

3

x

x

x

4

63

x

x

)2)(2(

)2(3

xx

x22 2)(

63

x

x

Rationalizing Practice

12 xx

2

51 x

Example 1

• Find the limit on the right. – First rationalize.– Then plug in.

Trigonometric Limits

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

Example 2

Evaluate the following limit:

Trig Identities

• Other trig limits may require you to apply trigonometric identities. Know the ones below.

Reteaching #1

• Evaluate the following limit

Example 3

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 1.1.3.

• Now let’s go over tan, cot, sec, csc.

y = tan x

• Note: the vertical lines are asymptotes. They are not part of the graph.

y = cot x

y = sec x

y = csc x

Example 4

• Use trig identities to simplify.

• Use one of your graphs to determine.

x

xx cos

cotlim

0

Homework

• Textbook 1a,c; 2a; 3a,c,d