Question

The value of \(\lim_{\theta \to 0} \frac{\tan\theta}{\theta}\) is

• A. 0
• B. 1
• C. \(\infty\)
• D. None of these

Hint:

Remember: \(\sin x \sim x\), \(\tan x \sim x\) as \(x \to 0\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is a standard trigonometric limit problem. It addresses the behavior of the tangent function relative to the angle itself as the angle approaches zero.
Step 2: Detailed Explanation:
Method 1: Using L'Hôpital's Rule (as it is a 0/0 form): \[ \lim_{\theta \to 0} \frac{\tan \theta}{\theta} = \lim_{\theta \to 0} \frac{\frac{d}{d\theta}(\tan \theta)}{\frac{d}{d\theta}(\theta)} \] \[ = \lim_{\theta \to 0} \frac{\sec^{2} \theta}{1} \] Substituting \( \theta = 0 \): \[ = \sec^{2}(0) = 1^{2} = 1 \] Method 2: Using basic limits: \[ \frac{\tan \theta}{\theta} = \frac{\sin \theta}{\theta \cos \theta} = \left( \frac{\sin \theta}{\theta} \right) \cdot \frac{1}{\cos \theta} \] We know \( \lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1 \) and \( \lim_{\theta \to 0} \cos \theta = 1 \). So, \( \lim_{\theta \to 0} \frac{\tan \theta}{\theta} = 1 \times \frac{1}{1} = 1 \).
Step 3: Final Answer:
The value of the limit is 1.