Understanding Asymptotic Behavior and Horizontal Asymptotes

The video tutorial explores the behavior of a function f(x) as x approaches negative infinity. It demonstrates how to rewrite the function for easier analysis by dividing the numerator and denominator by the highest degree term. The tutorial explains the concept of horizontal asymptotes and provides examples to illustrate how functions approach certain values as x becomes very large or very negative. The video concludes with a final example, reinforcing the understanding of function behavior and asymptotes.

What is the purpose of dividing both the numerator and the denominator by the highest degree term in the denominator?

• To increase the degree of the numerator

• To eliminate the denominator

• To change the function's value

• To simplify the expression for easier analysis

As x becomes very negative, what happens to the term 5/x in the function?

• It becomes a large positive number

• It approaches zero

• It becomes a large negative number

• It remains constant

How does the function f(x) behave as x approaches negative infinity?

• It oscillates

• It remains constant

• It approaches positive infinity

• It approaches negative infinity

What is a horizontal asymptote?

• A vertical line the graph approaches

• A point where the graph intersects the x-axis

• A line the graph never crosses

• A horizontal line the graph approaches as x becomes very large or very small

How can you determine the horizontal asymptote of a function?

• By finding the roots of the numerator

• By dividing all terms by the highest degree term in the denominator

• By setting the function equal to zero

• By finding the derivative of the function

What happens to the function q(x) as x approaches infinity?

• It approaches a vertical asymptote

• It approaches a constant value

• It approaches zero

• It becomes undefined

When analyzing end behavior, why can lower degree terms be ignored?

• They are always zero

• They have no impact on the function

• They grow faster than higher degree terms

• They are overwhelmed by higher degree terms

What is the horizontal asymptote of the function f(x) as x approaches negative infinity?

• y = -3

• y = infinity

• y = 0

• y = 3

What is the significance of the highest degree term in determining end behavior?

• It determines the y-intercept

• It dictates the function's growth rate

• It determines the x-intercept

• It is irrelevant to end behavior

What is the result of dividing 3x^4 by x^4 in the context of end behavior?

• It becomes undefined

• It becomes 3

• It becomes x

• It becomes zero