Understanding Limits through Factoring

The video tutorial explains how factoring can be used to determine limits of rational functions. It covers two examples: one where the limit is evaluated as X approaches 2 and another as X approaches -2. In both cases, the function is factored to simplify the expression, allowing for direct substitution to find the limit. The tutorial emphasizes the importance of recognizing holes and discontinuities in the graph of a function and how these affect the calculation of limits.

Why is factoring important when determining limits of rational functions?

• It changes the function to a polynomial.

• It makes the function continuous everywhere.

• It eliminates all discontinuities.

• It helps in simplifying the function for direct substitution.

What is the issue with performing direct substitution for the limit as x approaches 2 in the first example?

• The function is continuous at x = 2.

• The denominator becomes zero.

• The numerator becomes zero.

• The function is undefined for all x.

What is the simplified form of the function in the first example after factoring?

• (x + 1) / (x + 3)

• (x - 2) / (x - 3)

• (x + 2) / (x + 4)

• (x - 1) / (x - 3)

What is the limit of the function as x approaches 2 in the first example?

• 5/3

• 2/3

• 3/5

• 1/2

In the second example, what is the graphical behavior of the function at x = -2?

• The function is continuous.

• The function has a hole.

• The function has a vertical asymptote.

• The function is undefined everywhere.

What is the algebraic form of the numerator in the second example after factoring?

• x^2 - 4x + 4

• x^2 - 2x + 4

• x^2 + 2x + 4

• x^2 + 4x + 4

What is the limit of the function as x approaches -2 in the second example?

• 14

• 12

• 10

• 8

What is the common factor in the numerator and denominator in the second example?

• x - 3

• x + 2

• x - 2

• x + 3

Why is it necessary to factor the numerator and denominator in the second example?

• To find the derivative.

• To simplify the function for direct substitution.

• To change the function to a polynomial.

• To eliminate all discontinuities.

What is the purpose of simplifying the function in both examples?

• To find the exact value of the limit.

• To eliminate all asymptotes.

• To make the function continuous everywhere.

• To change the function to a polynomial.