3
QuizMultiple Choice

Limits, Asymptotes, and End Behavior — Quiz

Question 3 • Precalculus

Which statement correctly identifies an asymptote of f of x is equal to the fraction with numerator 16 x squared minus 35 and denominator x squared minus 5 using limits?

Answer
A
Because limit x to plus or minus infinity of f times x is equal to 7 , the function has a horizontal asymptote at y is equal to 7 .
B
Because limit x to plus or minus infinity of f times x is equal to 16 , the function has a vertical asymptote at x is equal to 16 .
C
Because limit x to plus or minus infinity of f times x is equal to 16 , the function has a horizontal asymptote at y is equal to 16 .
D
Because limit x to plus or minus infinity of f times x is equal to 7 , the function has a vertical asymptote at x is equal to 7 .

Explanation

As x approaches positive or negative infinity, the leading terms dominate, so f(x) approaches 16x²/x² = 16. A finite limit as x approaches infinity identifies a horizontal asymptote, so the asymptote is y = 16. Therefore, choice C is correct.

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