The following table provides approximate numerical values (rounded to three…

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

Problem

The following table provides approximate numerical values (rounded to three significant figures) for five different functions evaluated at three large positive arguments.

x

\ln x

\sqrt{x}

x^{\frac{1}{3}}

x

e^{0.1x}

------------------------------------------------- 10 2.30 3.16 2.15 10 2.72 100 4.61 10.0 4.64 100

2.20 \times 10^{4}

500 6.21 22.4 7.94 500

1.48 \times 10^{21}

The tabulated data suggest distinct long-run growth behaviors for the five functions listed. (a) Based on the data, rank the five functions in order of increasing growth rate as

x \to \infty

. Provide your ranking from smallest to largest. (b) For each pair of consecutive functions in your ranking, evaluate the limit of their quotient as

x \to \infty

. Show all work, and explain how each limit supports the ordering you obtained in part (a). (c) Consider the quotient

Q(x)=\frac{\sqrt{x}}{e^{0.1x}}

. Determine

\displaystyle \lim_{x \to \infty} Q(x)

and interpret the result in terms of a horizontal asymptote for the graph of

y=Q(x)

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