1
QuizMultiple Choice

Limits, Asymptotes, and End Behavior — Quiz

Question 1 • Precalculus

Consider f of x is equal to the fraction with numerator 7 x squared plus x plus 1 and denominator x to the 4th power plus 1 Which statement correctly uses limits to determine the end behavior of f of x ?

Answer
A
limit x to plus or minus infinity of the fraction with numerator 7 x squared plus x plus 1 and denominator x to the 4th power plus 1 is equal to limit x to plus or minus infinity of the fraction with numerator 7 x squared and denominator 1 , so the end behavior of the function is that as x to plus or minus infinity , f of x rightarrow infinity .
B
limit x to plus or minus infinity of the fraction with numerator 7 x squared plus x plus 1 and denominator x to the 4th power plus 1 is equal to limit x to plus or minus infinity of the fraction with numerator 7 and denominator x squared , so the end behavior of the function is that as x to plus or minus infinity , f of x rightarrow 0 .
C
limit x to plus or minus infinity of the fraction with numerator 7 x squared plus x plus 1 and denominator x to the 4th power plus 1 is equal to limit x to plus or minus infinity of 1 over 1 , so the end behavior of the function is that as x to plus or minus infinity , f of x rightarrow 1 .
D
limit x to plus or minus infinity of the fraction with numerator 7 x squared plus x plus 1 and denominator x to the 4th power plus 1 is equal to limit x to plus or minus infinity of the fraction with numerator 7 and denominator x squared , so the end behavior of the function is that as x to plus or minus infinity , f of x rightarrow 7 .

Explanation

The denominator has degree 4, while the numerator has degree 2, so the fraction approaches 0 as x grows in either direction. Dividing the numerator and denominator by x⁴ gives (7/x² + 1/x³ + 1/x⁴)/(1 + 1/x⁴), whose limit is 0. Choice B expresses this end behavior; the function does not approach 7 or infinity.

AI-written and checked against the verified answer.

Next