Which of the following best explains why the graph of has a horizontal asymptote at ?

MATHEMATICS · Single correct · AP · AP Precalculus · Polynomial and Rational Functions · Polynomial and Rational Functions

Problem

Which of the following best explains why the graph of

h(x)=\dfrac{3x^{2}-7}{5x^{2}+4x+1}

has a horizontal asymptote at

y=\dfrac{3}{5}

? (a) Because the denominator eventually grows faster than the numerator,

h(x)\to 0

|x|\to\infty

, giving a horizontal asymptote of

y=0

. (b) Because canceling the

x^{2}

terms and comparing the remaining coefficients leaves

\dfrac{5}{3}

, so the horizontal asymptote is

y=\dfrac{5}{3}

. (c) Because the leading terms

3x^{2}

and

5x^{2}

dominate for large

|x|

, so the function approaches the ratio of their coefficients. (d) Because the linear term

4x

in the denominator eventually dominates the numerator, shifting the end behavior to

y=\dfrac{4}{5}