Chapter 9.2, Problem 29E

(a).

To determine

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

Answer to Problem 29E

The value of $\underset{\theta \to 0}{\lim }\frac{\sin 3\theta }{\sin 4\theta }$ is $\frac{3}{4}$

Explanation of Solution

Given Information: The limit to be calculated is $\underset{\theta \to 0}{\lim }\frac{\sin 3\theta }{\sin 4\theta }$

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.

  $\underset{\_}{\text{L'H<mover accent="true"><mo> o</mo> <mo>^</mo></mover> pital's Rule}}\text{: If }f(a)=g(a)=0,\text{ then }\underset{x\to a}{\lim }\frac{f(x)}{g(x)}=\underset{x\to a}{\lim }\frac{f^{\prime }(x)}{g^{\prime }(x)}$

Calculation:

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

  $\begin{array}{cc}\left[\theta \right]& \left[\frac{sin3\theta }{sin4\theta }\right]\\ {\text{10}}^0& \text{-0}\text{.18647}\\ {10}^{-1}& \text{0}\text{.75888}\\ {10}^{-2}& \text{0}\text{.75009}\\ {10}^{-3}& \text{0}\text{.75}\\ {10}^{-4}& \text{0}\text{.75}\end{array}$

In the above table, as $\theta$ approaches $0$ , the value of the expression approaches $0.75$ .

Therefore, the estimated value of the limit is $0.75$

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

  $\underset{\theta \to 0}{\lim }\frac{\sin 3\theta }{\sin 4\theta }=\underset{\theta \to 0}{\lim }\frac{3\cos 3\theta }{4\cos 4\theta }$

  $=\frac{3}{4}\underset{\theta \to 0}{\lim }\frac{\cos 3\theta }{\cos 4\theta }$

  $=\frac{3}{4}\cdot \frac{\cos 0}{\cos 0}$

  $=\frac{3}{4}$

Therefore, the limit is $\frac{3}{4}\left(=0.75\right)$ and it is indeed the same value asestimated earlierusing tables.