Understanding Limits of Rational Functions

The video tutorial explains how to find the limit of a rational function as x approaches -1. Initially, direct substitution results in division by zero, so the function is simplified by determining if x + 1 is a factor of the numerator using synthetic division. After confirming x + 1 is a factor, the function is simplified, allowing the limit to be found through direct substitution. The limit is calculated to be -5. The tutorial concludes with a graphical verification, showing the function's graph with a hole at x = -1, confirming the limit is -5.

What is the main challenge when finding the limit of a rational function as x approaches a value that makes the denominator zero?

• The numerator becomes zero.

• The function becomes undefined due to division by zero.

• The function becomes infinite.

• The function becomes a constant.

What is the first step in simplifying a rational function to find its limit?

• Determining if a factor of the denominator is also a factor of the numerator.

• Finding the derivative.

• Performing direct substitution.

• Graphing the function.

Which method can be used to determine if x + 1 is a factor of the numerator?

• Synthetic division

• Partial fraction decomposition

• Integration

• Completing the square

In synthetic division, what is the constant used when dividing by x + 1?

• 1

• -1

• 2

• 0

What does a remainder of zero indicate in synthetic division?

• The divisor is not a factor.

• The divisor is a factor.

• The function is undefined.

• The function has a vertical asymptote.

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

• Finding the derivative.

• Graphing the function.

• Using the quadratic formula.

• Performing direct substitution.

What does the common factor between the numerator and denominator represent graphically?

• A vertical asymptote

• A horizontal asymptote

• A hole in the graph

• A point of intersection

What is the limit of the function as x approaches -1 after simplification?

• 0

• -1

• -5

• 5

What shape does the graph of the simplified function take?

• A line

• A circle

• A parabola

• An ellipse

What is observed as x approaches -1 from both sides on the graph?

• The function value remains constant.

• The function value approaches -5.

• The function value approaches infinity.

• The function value approaches 0.