Limits by Factoring and the Squeeze Theorem

Lesson 1.1.10

Learning Objectives

• Given a rational function, evaluate the limit as xc by factoring the numerator and/or denominator.

• Given a function, evaluate the limit as xc using the Squeeze Theorem.

Review of Factoring

• For this lesson, you need to know all types of factoring from Algebra II/Pre-calc

• This includes:– GCF factoring– Quadratic trinomial factoring– Difference of squares– Sum/difference of cubes– Synthetic division

GCF Factoring

• x3 – 5x

• 2x + 16

Quadratic Trinomial Factoring

• x2 + 5x + 6

• x2 – 5x + 4

• x2 + 3x – 4

• x2 – 7x – 30

Difference of Squares

a2 – b2 factors to (a + b)(a – b)• x2 – 4

• x2 – 9

• 9x2 – 25

• 4x2 – 49

Sum/Difference of Cubes

a3 + b3 factors to

(a + b)(a2 – ab + b2)

• x3 + 27

• 8x3 + 125

a3 – b3 factors to

(a – b)(a2 + ab + b2)

• x3 – 125

• 27x3 – 8

Synthetic Division

• This will be useful for polynomials of degree 3 or higher

x3 + 4x2 + x – 6

• Typically, you would use p/q to figure out which roots to try.

• The c value of the limit, however, will likely be one of the roots. Try that first.

Factor x3 + 4x2 + x – 6. c value is -2

Evaluating Limits by Factoring

• We can evaluate a limit by factoring the numerator and/or denominator and “canceling out” like factors.

• After canceling out, we just plug in the c value.

Example 1

• Evaluate the following limit.

Example 2

• Evaluate:

Example 3

x

xx

3

0lim

Example 4

1lim

3

1

x

xxx

Example 5

4

64lim

2

23

2

x

xxxx

Factoring Wrap-Up

• Thought question: Do you believe that the two functions on the right are equal? Why or why not?

5)(5

25)(

2

xxgx

xxf

Squeeze Theorem

• Suppose the function f(x) is in between two other functions, g(x) and h(x)

• Not only that, but as the limit as xc for g(x) and h(x) is equal.

• As a result, the limit of f(x) as xc must be the same as for g(x) and h(x).

More Formally…

Sometimes, this Squeeze Theorem is known as the Sandwich Theorem.

Example 6

Use the Squeeze Theorem to find

given

Applying the Squeeze Theorem

• Suppose that you were told to evaluate the limit below.

• Calculating this limit on its own is very difficult.

• This function, however, is “squeezed” in between two functions with the same limit. This limit is easy to find.

• Remember: the cosine of any angle must be in between -1 and 1.

• Thus, when you multiply any number by a cosine, its magnitude becomes smaller.

• Therefore, we can make the following argument:

xx

xx )1

cos(

Visually

Therefore…

• Let’s instead take the limits of -|x| and |x| as x0

• You can just plug in 0 for x.

• You will get a limit of 0 for both functions.

• Because x*cos(1/x) is in between these two functions, and both have limits of 0, x*cos(1/x) must also have a limit of 0 as x0.

Wrap-Up

• You can use factoring to determine fraction limits algebraically.

• You can find the limit of a function in between two functions using the Squeeze Theorem.

Homework

• Textbook 1a-d, 2