Understanding Limits of Rational Functions

The video tutorial explains how to find the limit of a rational function as a variable approaches zero. It covers two algebraic methods: factoring and expansion, to simplify the function and find the limit. Both methods reveal a common factor that results in a hole in the graph, which does not affect the limit. The tutorial concludes by verifying the limit graphically, showing that the function approaches a value of 4 as the variable approaches zero.

What is the main challenge when trying to find the limit of a rational function using direct substitution?

• The function becomes undefined due to division by zero.

• The function becomes too complex to solve.

• The function results in an infinite limit.

• The function has no real solutions.

Which technique involves recognizing the numerator as a difference of squares?

• Substitution

• Graphical Analysis

• Factoring

• Expansion

What is the result of simplifying the expression using the factoring technique?

• A limit of 4

• A limit of 2

• A limit of 6

• A limit of 0

What is the alternative method to factoring for finding the limit?

• Expansion

• Direct Substitution

• Numerical Approximation

• Graphical Analysis

In the expansion method, what is the first step after expanding the numerator?

• Simplifying the denominator

• Multiplying the binomials

• Performing direct substitution

• Factoring out common terms

What common factor is found in both the numerator and denominator in the expansion method?

• H squared

• 2

• 4

• H

What does the presence of a common factor of H in the numerator and denominator indicate graphically?

• A point of discontinuity

• A vertical asymptote

• A hole in the graph

• A horizontal asymptote

What is the final result of the limit using the expansion method?

• 6

• 4

• 2

• 0

What does the graph of the function verify about the limit?

• The limit is 4

• The limit is 2

• The limit is 0

• The limit is infinite

What does the hole in the graph represent in terms of the limit?

• A point where the limit does not exist

• A point that affects the limit

• A point of maximum value

• A point that does not affect the limit