Chapter 9.2, Problem 32E

(a).

To determine

To Estimate: The value of a given limit using tables and verify it with the value of the limit obtained using L'Hôpital's Rule.

Answer to Problem 32E

The value of $\underset{x\to \infty }{\lim }\frac{x-2x^2}{3x^2+5x}$ is $-0.66667$

Explanation of Solution

Given Information: The limit to be calculated is $\underset{x\to \infty }{\lim }\frac{x-2x^2}{3x^2+5x}$

Method Used: Calculate the value of the expression for suitable values of $x$ and observe the trend to estimate the limit. Then compare it with the value obtained using L'Hôpital's Rule.

  $\underset{\_}{\text{L'H<mover accent="true"><mo> o</mo> <mo>^</mo></mover> pital's Rule}}\text{: If }f(x)\text{ and }g(x)\text{ approach infinity as }x\to a,\text{ then }\underset{x\to a}{\lim }\frac{f(x)}{g(x)}=\underset{x\to a}{\lim }\frac{f^{\prime }(x)}{g^{\prime }(x)}$

Calculation:

(i). The values of the expression at various values of $x$ are

  $\begin{array}{l}\begin{array}{cc}x& \frac{x-2x^2}{3x^2+5x}\\ {\text{10}}^0& \text{-0}\text{.125}\\ {\text{10}}^1& \text{-0}\text{.54286}\\ {\text{10}}^2& \text{-0}\text{.65246}\\ {\text{10}}^3& \text{-0}\text{.66522}\\ {\text{10}}^4& \text{-0}\text{.66652}\end{array}\\ \begin{array}{cc}{\text{10}}^5& \text{-0}\text{.66665}\\ {\text{10}}^6& \text{-0}\text{.66667}\\ {\text{10}}^7& \text{-0}\text{.66667}\end{array}\end{array}$

In the above table, as $x$ approaches $\infty$ , the value of the expression approaches $-0.66667$ .

Therefore, the estimated value of the limit is $-0.66667$

(ii). Now apply L'Hôpital's Rule to write

  $\underset{x\to \infty }{\lim }\frac{x-2x^2}{3x^2+5x}=\underset{x\to \infty }{\lim }\frac{1-4x}{6x+5}$

  $=\underset{x\to \infty }{\lim }\left(\frac{0-4}{6+0}\right)\left[\text{L'Hôpital's Rule applied again as limit is still in }\frac{\infty }{\infty }\text{ form}\right]$

  $=\underset{x\to \infty }{\lim }\frac{-4}{6}$

  $=\frac{-2}{3}$

Therefore, the limit is $\frac{-2}{3}(=-0.6\overline{6})$ and it is indeed approximately the same value as estimated earlier using tables.