Question

in exercises 35–38, only one of the following is true.
i. ( f ) grows faster than ( g ).
ii. ( g ) grows faster than ( f ).
iii. ( f ) and ( g ) grow at the same rate.
use the given graph of ( f/g ) to determine which one is true.
35.
0, 100 by -1000, 10000
36.
0, 10 by -0.5, 1
37.
0, 100 by -1, 1.5
38.
0, 20 by -1, 3

Explanation:

Problem 35

Step1: Analyze the graph of \( \frac{f}{g} \)

The graph of \( \frac{f}{g} \) on \([0, 100]\) is increasing and going to infinity as \( x \) increases.

Step2: Relate to growth rates

If \( \lim_{x \to \infty} \frac{f(x)}{g(x)} = \infty \), then \( f \) grows faster than \( g \). Since the graph of \( \frac{f}{g} \) is increasing without bound, \( \frac{f}{g} \) tends to infinity as \( x \to \infty \), so \( f \) grows faster than \( g \).

Step1: Analyze the graph of \( \frac{f}{g} \)

The graph of \( \frac{f}{g} \) on \([0, 10]\) approaches 0 as \( x \) increases.

Step2: Relate to growth rates

If \( \lim_{x \to \infty} \frac{f(x)}{g(x)} = 0 \), then \( g \) grows faster than \( f \). Since the graph of \( \frac{f}{g} \) tends to 0 as \( x \to \infty \), \( g \) grows faster than \( f \).

Step1: Analyze the graph of \( \frac{f}{g} \)

The graph of \( \frac{f}{g} \) on \([0, 100]\) approaches a non - zero, finite limit (a constant) as \( x \) increases.

Step2: Relate to growth rates

If \( \lim_{x \to \infty} \frac{f(x)}{g(x)} = L\) where \( L\) is a non - zero finite number, then \( f \) and \( g \) grow at the same rate. Since the graph of \( \frac{f}{g} \) approaches a constant, \( f \) and \( g \) grow at the same rate.

Answer:

i. \( f \) grows faster than \( g \)

Problem 36