Which of the following statements best describes the value of ?

MATHEMATICS · Single correct · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Which of the following statements best describes the value of

\displaystyle \lim_{x\to\infty}\frac{x^{5}}{e^{x}}

? (a) The limit equals

\infty

because a fifth–degree polynomial will eventually outgrow an exponential function. (b) The limit equals

0

because an exponential function eventually outgrows any power function. (c) The limit equals

1

because the leading powers in the numerator and denominator "cancel" for large

x

.},{ (d) The limit equals

120

after five applications of L’Hospital’s Rule give a constant numerator over

e^{x}

.}],