Chapter 11.4, Problem 42E

a.

To determine

To complete: the table and numerically estimate the limit as x approaches infinity for the function.

Answer to Problem 42E

  $\underset{\mathrm{x}\to \mathrm{\infty }}{\lim }\mathrm{f}(\mathrm{x})=2.$

Explanation of Solution

Given information: Given function is $\mathrm{f}(\mathrm{x})=\frac{2\sqrt{{\mathrm{x}}^2+9000}}{\mathrm{x}}$ and given table is;

x	${10}^0$	${10}^1$	${10}^2$	${10}^3$	${10}^4$	${10}^5$	${10}^6$
f(x)

  $$

Calculation:

Here, plug the value of x into the given function to complete the table:

x	${10}^0$	${10}^1$	${10}^2$	${10}^3$	${10}^4$	${10}^5$	${10}^6$
f(x)	$189.7472$	$19.0788$	$2.7568$	$2.0090$	$2.0000$	$2.0000$	$2.0000$

See that f(x) approaches 2 as x $\to \mathrm{\infty }$ , so estimate that;

  $\underset{\mathrm{x}\to \mathrm{\infty }}{\lim }\mathrm{f}(\mathrm{x})=2.$

b.

To determine

To graph: the function and estimate the limit graphically.

Answer to Problem 42E

From the graph, $\underset{\mathrm{x}\to \mathrm{\infty }}{\lim }\mathrm{f}(\mathrm{x})=2$ .

Explanation of Solution

Given information: $\mathrm{f}(\mathrm{x})=\frac{2\sqrt{{\mathrm{x}}^2+9000}}{\mathrm{x}}$ .

Calculation:

The graph of the function is shown below.

  

From the graph, see that y = 2 seems to be horizontal asymptote, so since the limit to $\pm \mathrm{\infty }$ of a function gives the horizontal asymptote of that function, estimate from looking at the graph $\underset{\mathrm{x}\to \mathrm{\infty }}{\lim }\mathrm{f}(\mathrm{x})=2$ .