Understanding Limits at Infinity of Rational Functions

The video tutorial explains how to determine the limit of a rational function as X approaches positive infinity. It discusses the importance of comparing the degrees of the numerator and denominator to predict the limit's outcome. If the numerator's degree is less than the denominator's, the limit is zero. The tutorial also provides a guideline for simplifying terms by dividing by the highest power of X in the denominator. It concludes with a graphical representation to confirm the limit and highlights the horizontal asymptote.

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

• Evaluate the function at X equals zero.

• Find the derivative of the function.

• Determine the degree of the numerator and denominator.

• Compare the coefficients of the terms.

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?

• Zero

• Infinity

• One

• Does not exist

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

• The limit is zero.

• The limit is the ratio of the leading coefficients.

• The limit is infinity.

• The limit does not exist.

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

• Add the highest power of X to each term.

• 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.

When simplifying the expression by dividing each term by the highest power of X in the denominator, what should you do next?

• Add a constant to each term.

• Multiply the terms back together.

• Evaluate the limit directly.

• Simplify each term individually.

According to the definition, what happens to a fraction where the numerator is a constant and the denominator increases without bound?

• The fraction remains constant.

• The fraction approaches infinity.

• The fraction approaches zero.

• The fraction does not exist.

What is the behavior of the function f(x) = 1/x as X approaches positive infinity?

• The function remains constant.

• The function approaches zero.

• The function approaches infinity.

• The function does not exist.

What is the significance of a horizontal asymptote in the context of limits at infinity?

• It represents the value the function approaches as X goes to infinity.

• It shows the function will oscillate indefinitely.

• It indicates the function will eventually reach infinity.

• It means the function will never cross the X-axis.

How can graphing a rational function help confirm the limit as X approaches infinity?

• By illustrating the function's end behavior and asymptotes.

• By showing the function's behavior near the origin.

• By displaying the function's derivative.

• By highlighting the function's maximum and minimum points.

What does it mean if a rational function has a horizontal asymptote at y = 0?

• The function's values increase indefinitely.

• The function will never be zero.

• The function's values approach zero as X goes to infinity.

• The function will eventually reach zero.