Chapter 11.4, Problem 40E

a.

To determine

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

Answer to Problem 40E

  $\underset{\mathrm{x}\to \mathrm{\infty }}{\lim }\mathrm{f}(\mathrm{x})=0.5$ .

Explanation of Solution

Given information: Given function is $\mathrm{f}(\mathrm{x})=4(4\mathrm{x}-\sqrt{16{\mathrm{x}}^2-\mathrm{x}})$ 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)	$0.508067$	$0.500784$	$0.500078$	$0.500008$	$0.500001$	$0.5$	$0.5$

See that f(x) approaches 0.5 as x $\to \mathrm{\infty }$ , so estimate that;

  $\underset{\mathrm{x}\to \mathrm{\infty }}{\lim }\mathrm{f}(\mathrm{x})=\mathrm{0.5.}$

b.

To determine

To graph: the function and estimate the limit graphically.

Answer to Problem 40E

From the graph, $\underset{\mathrm{x}\to \mathrm{\infty }}{\lim }\mathrm{f}(\mathrm{x})=0.5$ .

Explanation of Solution

Given information: $\mathrm{f}(\mathrm{x})=4(4\mathrm{x}-\sqrt{16{\mathrm{x}}^2-\mathrm{x}})$ .

Calculation:

The graph of the function is shown below.

  

From the graph, see that y = 0.5 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 $\underset{\mathrm{x}\to \mathrm{\infty }}{\lim }\mathrm{f}(\mathrm{x})=0.5$ .