Chapter 11.4, Problem 38E

a.

To determine

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

Answer to Problem 38E

  $\underset{x\to \infty }{\lim }f(x)=0$ .

Explanation of Solution

Given information: Given function is $f(x)=3x-\sqrt{9x^2+1}$ 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)	$-1.62\times {10}^{-1}$	$-1.67\times {10}^{-2}$	$-1.67\times {10}^{-3}$	$-1.67\times {10}^{-4}$	$-1.67\times {10}^{-5}$	$-1.67\times {10}^{-6}$	$-1.67\times {10}^{-7}$

See that f(x) approaches 0 as x $\to \infty$ , so estimate that;

  $\underset{x\to \infty }{\lim }f(x)=0$

b.

To determine

To graph: the function and estimate the limit graphically.

Answer to Problem 38E

From the graph, $\underset{x\to \infty }{\lim }f(x)=0$ .

Explanation of Solution

Given information: $f(x)=3x-\sqrt{9x^2+1}$ .

Calculation:

The graph of the function is shown below.

  

From the graph, see that y = 0 seems to be horizontal asymptote, so since the limit to $\pm \infty$ of a function gives the horizontal asymptote of that function , estimate from looking at the graph $\underset{x\to \infty }{\lim }f(x)=0$ .