Understanding Limits at Infinity of Rational Functions

The video tutorial explains how to determine limits at infinity for rational functions. It introduces basic limits, such as when the numerator is a constant and the denominator increases or decreases without bound, leading the function value to approach zero. The tutorial provides shortcuts based on the degrees of the numerator and denominator to determine limits. It also outlines a guideline for finding limits at infinity by dividing each term by the highest power of x in the denominator. The video concludes with graph verification to illustrate the concepts.

What is the limit of a rational function where the numerator is a constant and the denominator increases without bound?

• The limit is 1

• The limit is 0

• The limit is infinity

• The limit does not exist

In the graph of f(x) = 1/x, what happens to the function value as x approaches negative infinity?

• It approaches infinity

• It approaches 1

• It approaches -1

• It approaches 0

If the degree of the numerator is less than the degree of the denominator, what is the limit of the rational function as x approaches infinity?

• The limit does not exist

• The limit is 0

• The limit is infinity

• The limit is 1

What is the limit of a rational function if the degrees of the numerator and denominator are equal?

• The limit is the ratio of the leading coefficients

• The limit is 0

• The limit is infinity

• The limit does not exist

What should you do to determine the limit of a rational function at infinity?

• Divide each term by the highest power of x in the denominator

• Multiply each term by the highest power of x in the numerator

• Subtract the highest power of x from each term

• Add the highest power of x to each term

When dividing each term by the highest power of x in the denominator, what happens to terms like 3/x as x approaches infinity?

• They approach 3

• They approach infinity

• They approach 0

• They remain unchanged

What is the limit of the rational function if the degree of the numerator is greater than the degree of the denominator?

• The limit is 0

• The limit does not exist

• The limit is the ratio of the leading coefficients

• The limit is infinity

In the graph of a rational function with a slant asymptote, what happens to the function values as x approaches positive infinity?

• They remain constant

• They oscillate

• They increase without bound

• They decrease without bound

What is a slant asymptote in the context of rational functions?

• A horizontal line the graph approaches

• A diagonal line the graph approaches

• A curve the graph approaches

• A vertical line the graph approaches

How does the graph verify the limits as x approaches positive and negative infinity?

• By showing zero function values

• By showing constant function values

• By showing increasing and decreasing function values without bound

• By showing oscillating function values