Consider the function for .

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Consider the function

y=(3x^{2}+5)e^{-x}

for

x\ge 0

. (i) Show that

\displaystyle \lim_{x\to\infty}(3x^{2}+5)e^{-x}=0

. (ii) Hence write the equation of any horizontal asymptote of the graph of

y=(3x^{2}+5)e^{-x}

. (iii) Determine the least integer

N

such that

|(3x^{2}+5)e^{-x}|<0.01

whenever

x\ge N

. Provide full justification of your answer.