Chapter 12.4, Problem 37E

(a)

To determine

The completed table and numerically estimatethe limit as approaches infinity.

Answer to Problem 37E

x	${10}^0$	${10}^1$	${10}^2$	${10}^3$	${10}^4$	${10}^5$	${10}^6$
f(x)	$-0.1622777$	$-0.0166620$	$-0.0016667$	$-0.0001667$	$-0.0000167$	$-0.0000017$	$-0.0000002$

Value of $\underset{x\to \infty }{\lim }\left(3\left(2x-\sqrt{4x^2+x}\right)\right)=-0.75$

Explanation of Solution

Given info:

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

  $f(x)=3\left(2x-\sqrt{4x^2+x}\right)$

Formula used:

If f(x) becomes arbitrarily close to a unique number L as approaches c fromeither side, then the limit of f(x) as approaches c is L. This is written as

  $\underset{x\to c}{\lim }f(x)=L$

Calculation:

We have

  $\begin{array}{l}f(x)=3\left(2x-\sqrt{4x^2+x}\right)\\ f({10}^0)=3\left(2\times {10}^0-\sqrt{4{\left({10}^0\right)}^2+{10}^0}\right)=-0.7082\\ f({10}^1)=3\left(2\times {10}^1-\sqrt{4{\left({10}^1\right)}^2+{10}^0}\right)=-0.7454\\ f({10}^2)=3\left(2\times {10}^2-\sqrt{4{\left({10}^2\right)}^2+{10}^0}\right)=-0.7495\\ f({10}^3)=3\left(2\times {10}^3-\sqrt{4{\left({10}^3\right)}^2+{10}^0}\right)=-0.7500\\ f({10}^4)=3\left(2\times {10}^4-\sqrt{4{\left({10}^4\right)}^2+{10}^0}\right)=-0.7500\\ f({10}^5)=3\left(2\times {10}^5-\sqrt{4{\left({10}^5\right)}^2+{10}^0}\right)=-0.7500\\ f({10}^6)=3\left(2\times {10}^6-\sqrt{4{\left({10}^6\right)}^2+{10}^0}\right)=-0.7500\end{array}$

Substituting directly,

x	${10}^0$	${10}^1$	${10}^2$	${10}^3$	${10}^4$	${10}^5$	${10}^6$
f(x)	$-0.7082$	$-0.7454$	$-0.7495$	$-0.7500$	$-0.7500$	$-0.7500$	$-0.7500$

Value of $\underset{x\to \infty }{\lim }\left(3\left(2x-\sqrt{4x^2+x}\right)\right)=-0.75$

Conclusion:

Thus,the value of $\underset{x\to \infty }{\lim }\left(3\left(2x-\sqrt{4x^2+x}\right)\right)=-0.75$

(b)

To determine

The graph of the function and estimate the limitgraphically.

Answer to Problem 37E

Value of $\underset{x\to \infty }{\lim }\left(3\left(2x-\sqrt{4x^2+x}\right)\right)=-0.75$

Explanation of Solution

Given info:

  $f(x)=3\left(2x-\sqrt{4x^2+x}\right)$

Formula used:

If f(x) becomes arbitrarily close to a unique number L as approaches c fromeither side, then the limit of f(x) as approaches c is L. This is written as

  $\underset{x\to c}{\lim }f(x)=L$

Calculation:

The graph is given by

  

Value of $\underset{x\to \infty }{\lim }\left(3\left(2x-\sqrt{4x^2+x}\right)\right)=-0.75$

Conclusion:

Thus,the value of $\underset{x\to \infty }{\lim }\left(3\left(2x-\sqrt{4x^2+x}\right)\right)=-0.75$