Mathematics

Problem

Answer the following questions related to the comparison of growth rates of common real-valued functions. a. (i) Define a horizontal asymptote of the graph of a function

y=f(x)

. b. (ii) For each pair of functions given below, determine which one grows faster as

x\to\infty

and justify your answer by evaluating an appropriate limit. (a)

f_1(x)=\log_{10}x

and

g_1(x)=x^{1/2}

(b)

f_2(x)=x^{3}

and

g_2(x)=2^{x}

c. (iii) Using your conclusions from part (ii), arrange the three functions

\log_{10}x,\;x^{3},\;2^{x}

in increasing order of their rates of growth as

x\to\infty

, clearly stating your reasoning.

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