Understanding Limits and Continuity in Rational Functions

The video tutorial explains how to find limits of a rational function at specific points. It covers the concept of continuity and discontinuity, using direct substitution for limits where the function is continuous. For points of discontinuity, the tutorial explores limits along different paths and uses graphical analysis to verify results. The first limit is evaluated using direct substitution, while the second limit involves analyzing paths and determining the behavior of the function as it approaches infinity.

What is a rational function continuous except for?

• Where both numerator and denominator are zero

• Everywhere

• Where the denominator is zero

• Where the numerator is zero

How can the limit be determined at a point where the function is continuous?

• By using integration

• By using the squeeze theorem

• By using direct substitution

• By using L'Hôpital's rule

What happens to the function at the point (2, 8) in the second limit?

• The function is continuous

• The function is undefined

• The function is zero

• The function is discontinuous

What is the result when the numerator approaches a constant and the denominator approaches zero?

• The limit is zero

• The limit is a constant

• The limit is infinite

• The limit does not exist

What does it mean if a limit approaches positive or negative infinity?

• The limit exists

• The limit does not exist

• The limit is a constant

• The limit is zero

What is the behavior of the function as it approaches the point (2, 8) from the positive side?

• The function remains constant

• The function approaches negative infinity

• The function approaches positive infinity

• The function approaches zero

What is the behavior of the function as it approaches the point (2, 8) from the negative side?

• The function approaches negative infinity

• The function approaches positive infinity

• The function approaches zero

• The function remains constant

What is the significance of one-sided limits in determining the existence of a limit?

• They simplify the limit

• They are irrelevant

• They show the limit does not exist

• They confirm the limit exists

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

• The function approaches infinity at discontinuities

• The function is zero at discontinuities

• The function has no discontinuities

• The function is continuous everywhere

How does the graph help in understanding the behavior of the function near discontinuities?

• It shows the function is continuous

• It shows the function approaches infinity

• It shows the function is undefined

• It shows the function is zero