Let be the function defined for by .

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

Problem

Let

f

be the function defined for

x>0

f(x)= x^3e^{-x}+5\ln x

. (a) Determine

\lim_{x\to\infty} x^3e^{-x}

. Show analytical justification. (b) Using your result from part (a), evaluate

\lim_{x\to\infty} f(x)

. (c) State the equation(s) of any horizontal asymptote(s) of the graph of

y=f(x)

. Justify your answer with limit notation. (d) Verify that your asymptote from part (c) satisfies the formal

\varepsilon

–

M

definition: for an arbitrary

\varepsilon>0

, find

M

such that

|f(x)-L|<\varepsilon

whenever

x>M

, where

L

is the limiting value found in part (b).