Understanding Limits at Infinity of Rational Functions

The video tutorial explains how to determine the limit of a rational function as X approaches infinity. It introduces a shortcut method based on the degrees of the numerator and denominator, and provides general guidelines for finding limits by dividing each term by the highest power of X in the denominator. The tutorial also discusses the use of graphs to confirm limits and understand horizontal asymptotes, emphasizing that a graph can cross a horizontal asymptote but not a vertical one.

What is the first step in determining the limit of a rational function as X approaches infinity?

• Identify the coefficients of the terms.

• Determine the degree of the numerator and denominator.

• Calculate the derivative.

• Graph the function.

If the degree of the numerator is less than the degree of the denominator, what is the limit of the rational function?

• Infinity

• Zero

• Does not exist

• The ratio of leading coefficients

When the degrees of the numerator and denominator are equal, how is the limit determined?

• By multiplying the coefficients

• By finding the ratio of the leading coefficients

• By subtracting the coefficients

• By adding the coefficients

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

• The limit does not exist

• The limit is zero

• The limit is infinity

• The limit is the ratio of leading coefficients

What is the general guideline for finding limits at infinity of rational functions?

• Multiply each term by the highest power of X

• Divide each term by the highest power of X in the numerator

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

• Subtract the highest power of X from each term

What is the result of dividing a constant by a term with X raised to a positive power as X approaches infinity?

• The constant itself

• Zero

• Infinity

• Negative infinity

In the example given, what is the simplified limit of the rational function?

• Infinity

• 0

• 5

• -5

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

• A line the graph approaches but never touches

• A point on the graph

• A line the graph crosses

• A vertical line on the graph

How can the limit of a rational function be confirmed graphically?

• By checking if the graph crosses the x-axis

• By observing the horizontal asymptote

• By finding the slope of the tangent

• By calculating the area under the curve

What is the horizontal asymptote of the function discussed in the video?

• Y = 0

• Y = 5

• Y = -5

• Y = -2.5