The functions ,

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

Problem

The functions

f

g

, and

h

are defined for all sufficiently large

x

f(x)=5x^3-7x^2+2,\qquad g(x)=e^{x},\qquad h(x)=x^3\ln x.

The end-behavior of ratios of these functions reveals how quickly one function grows relative to another. (a) Explain how the existence and value of

\displaystyle \lim_{x\to\infty}\frac{p(x)}{q(x)}

indicate the comparative growth rates of two positive functions

p

and

q

. Include each of the following cases in your discussion: (i) the limit equals

0

, (ii) the limit equals a non-zero finite constant, and (iii) the limit is infinite. (b) For each ordered pair

(p,q)

among

\{f,g,h\}

, evaluate

\displaystyle \lim_{x\to\infty}\frac{p(x)}{q(x)}

or explain why that limit is infinite. Provide clear reasoning rather than only a numeric answer. (c) Use the results from part (b) to rank

f

g

, and

h

from slowest to fastest growth as

x\to\infty

. Justify your ranking. (d) Determine all horizontal asymptotes of the six quotient functions

\dfrac{p(x)}{q(x)}

where

p

and

q

are distinct members of

\{f,g,h\}

. Justify each answer in a sentence that connects the limit you found to the definition of a horizontal asymptote.