Question

Evaluate the limit \( \lim_{\theta \to \frac{\pi}{2}} \frac{1 - \sin \theta}{\left(\frac{\pi}{2} - \theta\right)\cos \theta} \)

• A. \( -\frac{1}{2} \)
• B. \( -1 \)
• C. 1
• D. \( \frac{1}{2} \)

Hint:

To resolve indeterminate forms such as \( \frac{0}{0} \) in limits, L'Hopital's Rule is a powerful method, which involves differentiating the numerator and denominator separately and then evaluating the limit.

Answer 1

The Correct Option is D

Step 1: Direct substitution.
We first attempt to directly substitute \( \theta = \frac{\pi}{2} \) into the expression. We know that:
- \( \sin \frac{\pi}{2} = 1 \),
- \( \cos \frac{\pi}{2} = 0 \),
- \( \frac{\pi}{2} - \frac{\pi}{2} = 0 \).
Thus, the expression becomes \( \frac{0}{0} \), which is an indeterminate form.

Step 2: Apply L'Hopital's Rule.
Since the limit gives an indeterminate form, we apply L'Hopital's Rule. L'Hopital's Rule states that if we have an indeterminate form \( \frac{0}{0} \), we can differentiate the numerator and the denominator separately and then take the limit.

Step 3: Differentiate the numerator and denominator.
The numerator is \( 1 - \sin \theta \), and the denominator is \( (\frac{\pi}{2} - \theta) \cos \theta \).
The derivative of the numerator is:
\[ \frac{d}{d\theta} (1 - \sin \theta) = -\cos \theta \]
The denominator is a product, so we apply the product rule:
\[ \frac{d}{d\theta} \left[ (\frac{\pi}{2} - \theta) \cos \theta \right] = -\cos \theta + (\frac{\pi}{2} - \theta) (-\sin \theta) \] \[ = -\cos \theta - (\frac{\pi}{2} - \theta) \sin \theta \]

Step 4: Evaluate the limit.
Now, we substitute \( \theta = \frac{\pi}{2} \) into the differentiated numerator and denominator:
- The numerator becomes \( -\cos \frac{\pi}{2} = 0 \),
- The denominator becomes \( -\cos \frac{\pi}{2} - (\frac{\pi}{2} - \frac{\pi}{2}) \sin \frac{\pi}{2} = 0 - 0 = 0 \).
Since the expression is still indeterminate, we need to apply L'Hopital's Rule again.

Step 5: Apply L'Hopital's Rule again.
For the second derivatives:
\[ f''(\theta) = \sin \theta \quad \text{and} \quad g''(\theta) = 2 \sin \theta - (\frac{\pi}{2} - \theta) \cos \theta \]
Substituting \( \theta = \frac{\pi}{2} \) into the second derivatives:
- The numerator \( f''(\theta) = \sin \frac{\pi}{2} = 1 \),
- The denominator \( g''(\theta) = 2 \sin \frac{\pi}{2} - (\frac{\pi}{2} - \frac{\pi}{2}) \cos \frac{\pi}{2} = 2 - 0 = 2 \).
Now, the limit is:
\[ \frac{1}{2} \]

Step 6: Conclusion.
Therefore, the value of the limit is \( \frac{1}{2} \), and the correct answer is option (D).