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

Concept: \[ \lim_{\theta \to 0} \frac{\sin\theta}{\theta} = 1, \quad \lim_{\theta \to 0} \frac{\tan\theta}{\theta} = 1 \]

Step 1: Rewrite using identity. \[ \frac{\tan\theta}{\theta} = \frac{\sin\theta}{\theta} \cdot \frac{1}{\cos\theta} \]

Step 2: Apply limits. \[ \lim_{\theta \to 0} \frac{\sin\theta}{\theta} = 1, \quad \lim_{\theta \to 0} \cos\theta = 1 \Rightarrow \lim_{\theta \to 0} \frac{1}{\cos\theta} = 1 \]

Step 3: Final value. \[ \lim_{\theta \to 0} \frac{\tan\theta}{\theta} = 1 \times 1 = 1 \]