Consider the two functions and

MATHEMATICS · Multi-part · A Level · Mathematics · Exponentials and logarithms · Exponentials and logarithms

Problem

Consider the two functions

p(x)=\ln x

and

q(x)=x^{1/3}

, defined for

x>1

. (a) Using appropriate properties of logarithmic and power functions, describe which of the two functions grows faster as

x\to\infty

. Provide a clear verbal argument that does not rely on explicit differentiation. (b) Based on your reasoning in part (a), determine the value of the limit

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

and justify why the limit must equal that value. (c) State the equation of any horizontal asymptote of the graph of

y=\dfrac{\ln x}{x^{1/3}}

and explain, in words, how the result of part (b) supports your conclusion.

Related theory

Limits
Growth Models
Exponential and Logarithmic Functions
Functions

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