Limits by Factoring and the Squeeze Theorem Lesson

Learning Objectives Given a rational function, evaluate the limit as x  c by factoring the numerator and/or denominator. Given a function, evaluate the limit as x  c 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 x 3 – 5x 2x + 16

Quadratic Trinomial Factoring x 2 + 5x + 6 x 2 – 5x + 4 x 2 + 3x – 4 x 2 – 7x – 30

Difference of Squares a 2 – b 2 factors to (a + b)(a – b) x 2 – 4 x 2 – 9 9x 2 – 25 4x 2 – 49

Sum/Difference of Cubes a 3 + b 3 factors to (a + b)(a 2 – ab + b 2 ) x x a 3 – b 3 factors to (a – b)(a 2 + ab + b 2 ) x 3 – x 3 – 8

Synthetic Division This will be useful for polynomials of degree 3 or higher x 3 + 4x 2 + 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 x 3 + 4x 2 + 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

Example 4

Example 5

Factoring Wrap-Up Thought question: Do you believe that the two functions on the right are equal? Why or why not?

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 x  c for g(x) and h(x) is equal. As a result, the limit of f(x) as x  c 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:

Visually

Therefore… Let’s instead take the limits of -|x| and |x| as x  0 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 x  0.

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