Two differentiable functions

MATHEMATICS · Multi-part · CBSE · Mathematics · Power Play · Power Play

Problem

Two differentiable functions, ff and gg, are defined for all x≥0x\ge 0 by f(x)=e0.02x,g(x)=x3.f(x)=e^{0.02x},\qquad g(x)=x^{3}. A student claims that ff "eventually dominates" gg as xx increases. (a) Describe, in words, what the statement lim⁡x→∞f(x)g(x)=∞\displaystyle \lim_{x\to\infty}\frac{f(x)}{g(x)}=\infty conveys about the long–run relationship between ff and gg. (b) Without carrying out explicit limit calculations, provide a qualitative argument to determine which function ultimately attains larger values for sufficiently large xx. Address the concept of growth rate in your explanation. (c) Let h(x)=ln⁡(f(x)g(x))h(x)=\ln\bigl(\tfrac{f(x)}{g(x)}\bigr). Describe the end–behavior of hh as x→∞x\to\infty and interpret its meaning in the context of comparing ff and gg. Explain your reasoning.

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