Understanding Limits of Rational Functions

The video tutorial explains how to evaluate the limits of rational functions as x approaches infinity using two methods. The first method involves analyzing the degrees of the numerator and denominator to determine the limit. The second method involves dividing each term by the highest power of the variable in the denominator and simplifying the expression. The tutorial provides examples for each method and concludes with a comparison of the results obtained from both approaches.

What are the two methods discussed for evaluating limits of rational functions as x approaches infinity?

• Substitution and elimination

• Analyzing degrees and dividing by highest power

• Graphical analysis and numerical approximation

• Using derivatives and integrals

In the first method, if the degree of the denominator is greater than the numerator, what is the limit?

• Undefined

• One

• Zero

• Infinity

What is the limit when the degrees of the numerator and denominator are equal?

• The ratio of the leading coefficients

• Undefined

• Zero

• Infinity

If the degree of the numerator is greater than the denominator, what does the limit approach?

• Negative infinity

• One

• Infinity

• Zero

What is the first step in the second method for determining limits?

• Subtracting the highest power of x

• Dividing terms by the highest power of x in the denominator

• Multiplying by the highest power of x

• Adding the highest power of x

When simplifying using the second method, what happens to terms with x in the denominator as x approaches infinity?

• They approach zero

• They remain constant

• They approach infinity

• They become undefined

In the second method, what is the result of dividing a term by itself?

• Zero

• One

• Infinity

• Undefined

What is the simplified form of 3x^4 divided by x^5?

• 3x^4

• 3/x

• 3x^5

• 3x

What is the final limit for the first rational function using the second method?

• Zero

• One

• Infinity

• Undefined

What conclusion can be drawn about the limits of rational functions as x approaches infinity?

• They are always zero

• They depend on the degrees of the numerator and denominator

• They are always infinity

• They are always undefined