As , consider the functions and . Which of the following statements correctly…

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

Problem

As x→∞x\to\infty, consider the functions f(x)=ln⁡xf(x)=\ln x and g(x)=x1/3g(x)=x^{1/3}. Which of the following statements correctly describes their relative growth rates? (a) lim⁡x→∞ln⁡xx1/3=∞\displaystyle\lim_{x\to\infty}\frac{\ln x}{x^{1/3}}=\infty, so ff grows faster than gg. (b) lim⁡x→∞ln⁡xx1/3=1\displaystyle\lim_{x\to\infty}\frac{\ln x}{x^{1/3}}=1, so ff and gg grow at the same rate. (c) lim⁡x→∞ln⁡xx1/3=0\displaystyle\lim_{x\to\infty}\frac{\ln x}{x^{1/3}}=0, so gg grows faster than ff. (d) lim⁡x→∞ln⁡xx1/3=0\displaystyle\lim_{x\to\infty}\frac{\ln x}{x^{1/3}}=0, so ff grows faster than gg.

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