#### To determine

**(a)**

**To find:** We have to compare the rate of growth of fx=x0.1 and gx=lnx by graphing both of the functions in different viewing rectangles. We also find the value of *x,* when the function *f* finally surpass the graph of *g.*

#### Answer

see the graph

#### Explanation

**Calculation:**

Here are two graphs which compare the rate of growth of fx=x0.1 and gx=lnx.

In these graph, blue graph represents the function gx=lnx and green graph represents the function fx=x0.1.

This graph shows that for small values of *x,* the function fx=x0.1 grow slowly than the function gx=lnx.

**Final statement:** see the graph

This graph shows that after x=3.431×1015 the graph fx=x0.1 finally surpasses the graph of gx=lnx.

#### To determine

**(b)**

**To find:** We have to graph the function hx=lnxx0.1 in a viewing rectangle that displays the behavior of the function as x→∞.

#### Answer

#### Explanation

This graph does not show the exact behavior of the graph h(x) as x→∞.

So, we draw the graph for large values of x.

This graph shows that as x→∞,h(x)x→0.

#### To determine

**(c)**

**To find:** We have to find the number *N* such that if x<N then lnxx0.1<0.1.

#### Explanation

**Calculation:** To find the value of *N*, first we draw the graph of the function lnxx0.1.

The graph of the function is

This graph shows that when x>1.3×1028,lnxx0.1<0.1.

So, *N* should be greater than or equal to 1.3×1028.