4
QuizMultiple Choice

Limits, Asymptotes, and End Behavior — Quiz

Question 4 • Precalculus

Which statement describes whether the function f of x is equal to the fraction with numerator 8 times open paren x minus 1 close paren squared and denominator open paren x minus 8 close paren squared has a vertical asymptote at x is equal to 8 ?

Answer
A
There is not a vertical asymptote at x is equal to 8 because limit x to 8 of f times x does not exist.
B
There is a vertical asymptote at x is equal to 8 because limit x to 8 raised to the negative power of f times x is equal to infinity and limit x to 8 raised to the plus power of f times x is equal to infinity .
C
There is a vertical asymptote at x is equal to 8 because limit x to infinity of f times x is equal to 8 .
D
There is not a vertical asymptote at x is equal to 8 because limit x to 8 raised to the negative power of f times x and limit x to 8 raised to the plus power of f times x are not opposites.

Explanation

As x approaches 8, the numerator 8(x − 1)² approaches 8(7)² = 392, which is positive. The denominator (x − 8)² approaches 0 while staying positive on both sides of 8, so f(x) grows without bound from the left and the right. Therefore, x = 8 is a vertical asymptote; choice B correctly describes this.

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