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Section 2.2 : The Limit
3. For the function \(\displaystyle g\left( \theta\right) = \frac{{\sin \left( {7\theta } \right)}}{\theta }\) answer each of the following questions.
- Evaluate the function at the following values of \(\theta \) compute (accurate to at least 8 decimal places). Make sure your calculator is set to radians for the computations.
- 0.5
- 0.1
- 0.01
- 0.001
- 0.0001
- -0.5
- -0.1
- -0.01
- -0.001
- -0.0001
- Use the information from (a) to estimate the value of \(\displaystyle \mathop {\lim }\limits_{\theta\to \,0} \frac{{\sin \left( {7\theta } \right)}}{\theta }\).
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a Evaluate the function at the following values of \(\theta \) compute (accurate to at least 8 decimal places). Make sure your calculator is set to radians for the computations. Show Solution- 0.5
- 0.1
- 0.01
- 0.001
- 0.0001
- -0.5
- -0.1
- -0.01
- -0.001
- -0.0001
Here is a table of values of the function at the given points accurate to 8 decimal places.
| \(\theta\) | \(g(\theta)\) | \(\theta\) | \(g(\theta)\) |
|---|---|---|---|
| 0.5 | -0.70156646 | -0.5 | -0.70156646 |
| 0.1 | 6.44217687 | -0.1 | 6.44217687 |
| 0.01 | 6.99428473 | -0.01 | 6.99428473 |
| 0.001 | 6.99994283 | -0.001 | 6.99994283 |
| 0.0001 | 6.99999943 | -0.0001 | 6.99999943 |
b Use the information from (a) to estimate the value of \(\displaystyle \mathop {\lim }\limits_{\theta\to \,0} \frac{{\sin \left( {7\theta } \right)}}{\theta }\). Show Solution
From the table of values above it looks like we can estimate that,
\[\mathop {\lim }\limits_{\theta \to 0} \frac{{\sin \left( {7\theta } \right)}}{\theta } = 7\]