Calculus

### **Detailed Step-by-Step Solution (GATE AE 2019 Q.25 / AE Q.15)** **1. Intuitive Physical Setup & Given Parameters**: - Evaluate the indeterminate form $\frac{0}{0}$ of $\lim_{\theta \to 0} \frac{\theta - \sin\theta}{\theta^3}$. **2. Step-by-Step Mathematical Derivation**: 1. **Taylor Series Expansion of $\sin\theta$**: Around $\theta = 0$: $\sin\theta = \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \dots = \theta - \frac{\theta^3}{6} + \mathcal{O}(\theta^5)$ 2. **Substituting into Numerator**: $\theta - \sin\theta = \theta - \left(\theta - \frac{\theta^3}{6} + \dots\right) = \frac{\theta^3}{6} - \mathcal{O}(\theta^5)$ 3. **Evaluating the Limit**: $\lim_{\theta \to 0} \frac{\theta - \sin\theta}{\theta^3} = \lim_{\theta \to 0} \frac{\frac{\theta^3}{6}}{\theta^3} = \frac{1}{6} \approx 0.16667$ 4. Rounding to 2 decimal places: $L \approx 0.17$ Official acceptable range is **[0.16, 0.17]**. **3. Comprehensive Options & Distractor Analysis**: - **Question Type**: Numerical Answer Type (NAT). - **Official Range**: **0.16 to 0.17**. **4. Official Master Key & Exam Takeaway**: - **Official Answer**: **0.17** (Range: **0.16 to 0.17**) - **Exam Rule**: Series expansion $\sin\theta \approx \theta - \theta^3/6$ evaluates limits 10x faster than 3 successive L'Hôpital differentiations.