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 discussing direct substitution and the issue of indeterminate forms like zero divided by zero. The tutorial then covers factoring techniques to eliminate removable discontinuities, allowing for simplification of the rational function. Finally, it demonstrates how to evaluate the limit using the simplified function, providing both decimal and fractional results.

What is the primary method to evaluate the limit of a rational function when there is no discontinuity?

• Using L'Hôpital's Rule

• Using numerical approximation

• Performing direct substitution

• Applying the Squeeze Theorem

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

• Direct substitution is not possible

• The limit does not exist

• The function is continuous

• The function is differentiable

What is the first step in factoring the expression x^4 - 81?

• Identify it as a sum of cubes

• Recognize it as a difference of squares

• Apply the quadratic formula

• Use synthetic division

Which of the following is a factor of x^4 - 81?

• x - 81

• x^2 + 81

• x^2 - 9

• x + 9

What is the challenge in factoring 2x^2 + 3x - 27?

• It is already factored

• The leading coefficient is not 1

• It is a perfect square

• It is a sum of cubes

What happens when a common factor is canceled in a rational function?

• The function becomes continuous

• A hole is created in the function

• The function becomes undefined

• The limit does not exist

What is the significance of a hole in a rational function?

• It makes the function undefined

• It does not affect the limit

• It affects the limit

• It indicates a vertical asymptote

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

• Perform direct substitution

• Apply the Squeeze Theorem

• Use numerical approximation

• Use L'Hôpital's Rule

What is the result of the limit after substituting x = 3?

• Both 7.2 and 36/5

• 7.2

• 36/5

• None of the above

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

• Check for asymptotes

• Perform direct substitution

• Graph the function

• Use a calculator for verification