2
QuizMultiple Choice

Limits, Asymptotes, and End Behavior — Quiz

Question 2 • Precalculus

Which statement correctly interprets this limit? limit x to plus or minus infinity of the fraction with numerator 28 x minus 10 x squared and denominator 4 x squared minus 1

Answer
A
The function f of x is equal to the fraction with numerator 28 x minus 10 x squared and denominator 4 x squared minus 1 has a horizontal asymptote at y is equal to 7 .
B
The function f of x is equal to the fraction with numerator 28 x minus 10 x squared and denominator 4 x squared minus 1 has a vertical asymptote at x is equal to 7 .
C
The function f of x is equal to the fraction with numerator 28 x minus 10 x squared and denominator 4 x squared minus 1 has a horizontal asymptote at y is equal to negative 5 halves .
D
The function f of x is equal to the fraction with numerator 28 x minus 10 x squared and denominator 4 x squared minus 1 has a vertical asymptote at x is equal to negative 5 halves .

Explanation

For a rational function whose numerator and denominator have the same degree, the limit as x approaches positive or negative infinity is the ratio of the leading coefficients. Here that ratio is −10/4 = −5/2, so the function approaches −5/2 at both ends. This means it has a horizontal asymptote at y = −5/2, which is choice C.

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