Previous Year Questions (PYQ): Limits & Derivatives

Important Tips for Solving JEE PYQs

1. Recognize the Form Immediately: The first step in any limit problem is to substitute the value and identify the indeterminate form (e.g., $0/0$, $1^\infty$, $\infty-\infty$). This dictates your entire strategy.

2. Prioritize Methods for 0/0: For the $0/0$ form, your order of attack should generally be: a) Standard Limits/Algebraic Manipulation (fastest) b) Series Expansion (very powerful for $x \to 0$) c) L'Hôpital's Rule (reliable, but can be slow if repeated differentiation is needed).

3. Shortcut for $1^\infty$: Always use the formula $L = e^{\lim_{x \to a} g(x)[f(x)-1]}$ for limits of the form $\lim_{x \to a} f(x)^{g(x)}$. It's a significant time-saver.

4. Differentiability Check: To check if a function is differentiable, first check for continuity. If it's discontinuous, it's not differentiable. If it is continuous, then check if LHD = RHD at the point in question.

5. Simplify Before Differentiating: In derivative problems, especially those involving inverse trigonometric functions, look for opportunities to simplify using algebraic or trigonometric substitutions before you start differentiating.

6. Master the Chain Rule: A vast majority of differentiation problems in JEE involve the chain rule, often with multiple nested functions. Practice it until it becomes second nature.

7. Don't Fear Piecewise Functions: For questions on continuity and differentiability of piecewise functions, focus only on the 'boundary points'. The function's behavior at these specific points is what the question is almost always about.