Understanding Limits of Rational Functions

The video tutorial explains how to determine the limit of a rational function as x approaches a specific value. It begins by attempting direct substitution, which results in an indeterminate form. The tutorial then demonstrates how to factor both the numerator and denominator to simplify the function. By canceling common factors, the function is simplified, allowing for direct substitution to find the limit. The process involves recognizing the difference of cubes and squares, factoring them, and simplifying the expression to evaluate the limit successfully.

What is the first step in evaluating the limit of a rational function?

• Using the quadratic formula

• Finding the derivative

• Graphing the function

• Performing direct substitution

What does the indeterminate form 0/0 indicate when evaluating a limit?

• The function is undefined

• The function is continuous

• The limit does not exist

• Simplification is needed

Which formula is used to factor the expression x^3 - 27?

• Quadratic formula

• Sum of cubes

• Difference of squares

• Difference of cubes

What is the binomial factor obtained from x^3 - 27?

• x^2 - 3x + 9

• x^2 + 3x + 9

• x + 3

• x - 3

How is the expression x^2 - 9 factored?

• (x + 3)(x - 3)

• (x + 3)(x + 3)

• (x - 3)(x - 3)

• (x + 9)(x - 9)

What does a common factor in the numerator and denominator represent graphically?

• A hole in the graph

• A horizontal asymptote

• A point of intersection

• A vertical asymptote

After simplifying the rational function, what is the next step to find the limit?

• Using L'Hôpital's rule

• Performing direct substitution

• Graphing the function

• Finding the derivative

What is the result of substituting x = 3 into the simplified function?

• 4.5

• 3

• 6

• 9

What is the final limit of the rational function as x approaches 3?

• 3

• 6

• 4.5

• 9

What common factor do 27 and 6 share in the final calculation?

• 9

• 6

• 3

• 2