Understanding One-Sided Limits and Vertical Asymptotes

The video tutorial explains the relationship between one-sided limits and vertical asymptotes. It describes how a vertical line x = 'A' is a vertical asymptote if the limit of f(x) as x approaches 'A' from either side is infinite. The tutorial provides examples of determining vertical asymptotes at x = 0 and x = 3 using limits and tables of values. It also discusses how rational functions can be used to identify vertical asymptotes by examining the values that make the denominator zero but not the numerator.

What is a vertical asymptote in the context of one-sided limits?

• A horizontal line that the graph approaches but never touches

• A point where the function has a maximum value

• A vertical line where the function approaches infinity

• A point where the function crosses the x-axis

In the first example, what happens to the function as x approaches 0 from the left?

• The function approaches negative infinity

• The function approaches positive infinity

• The function remains constant

• The function oscillates

How can we verify the limit as x approaches 0 from the left?

• By using a graph

• By creating a table of values

• By using a calculator

• By solving an equation

What is the significance of the limit being equal to negative infinity in the first example?

• It indicates a horizontal asymptote

• It means the function is undefined

• It shows that the limit exists

• It confirms a vertical asymptote at x = 0

In the second example, what happens to the function as x approaches 3 from the right?

• The function remains constant

• The function decreases

• The function approaches positive infinity

• The function approaches negative infinity

What does the limit being equal to positive infinity indicate in the second example?

• The limit exists

• There is a vertical asymptote at x = 3

• The function is undefined

• The function has a maximum value

How can we confirm the vertical asymptote at x = 3?

• By solving an equation

• By using a calculator

• By creating a table of values

• By using a graph

What is the role of the denominator in identifying vertical asymptotes in rational functions?

• It must be zero for a vertical asymptote

• It determines the horizontal asymptote

• It has no role in vertical asymptotes

• It must be greater than the numerator

What condition must be met for a vertical asymptote to exist in a rational function?

• The function must be undefined

• Both numerator and denominator must be zero

• The denominator must be zero and not the numerator

• The numerator must be zero

Which values of x create vertical asymptotes in the given function?

• x = 1 and x = 3

• x = 1 and x = 2

• x = 0 and x = 3

• x = 2 and x = 4