Chapter 8.3, Problem 53E

(a)

To determine

To show: when f decreases faster than g then, $\left|f\right|$ grows faster than $\left|g\right|$ as $x\to \infty$

Answer to Problem 53E

  $\left|f\right|$ grows faster than $\left|g\right|$ as $x\to \infty$ is shown.

Explanation of Solution

Given:

  $f\left(x\right)$ and $g\left(x\right)$ becomes and remains negative as $x\to \infty$

Calculation:

When $f\left(x\right)$ grows faster than $g\left(x\right)$ then,

  $\begin{array}{l}\underset{x\to \infty }{\lim }\frac{\left|f\left(x\right)\right|}{\left|g\left(x\right)\right|}=\underset{x\to \infty }{\lim }\frac{-f\left(x\right)}{-g\left(x\right)}\\ \text{ =}\underset{x\to \infty }{\lim }\frac{f\left(x\right)}{g\left(x\right)}\\ \text{ = }\infty \end{array}$

Conclusion:

  $\left|f\right|$ grows faster than $\left|g\right|$ as $x\to \infty$ is shown.

(b)

To determine

when f and g decreases at same rate then, $\left|f\right|$ and $\left|g\right|$ grows at the same rate as $x\to \infty$

Answer to Problem 53E

  $\left|f\right|$ and $\left|g\right|$ grows at the same rate as $x\to \infty$ is shown.

Explanation of Solution

Given:

  $f\left(x\right)$ and $g\left(x\right)$ becomes and remains negative as $x\to \infty$

Calculation:

When both grows at same rate then,

  $\begin{array}{l}\underset{x\to \infty }{\lim }\frac{\left|f\left(x\right)\right|}{\left|g\left(x\right)\right|}=\underset{x\to \infty }{\lim }\frac{-f\left(x\right)}{-g\left(x\right)}\\ \text{ =}\underset{x\to \infty }{\lim }\frac{f\left(x\right)}{g\left(x\right)}\\ \ne \text{ 0}\end{array}$

Conclusion:

  $\left|f\right|$ and $\left|g\right|$ grows at the same rate as $x\to \infty$ is shown.