Limits and Derivatives introduces calculus to Indian students for the first time. It is one of the most formula-driven chapters in Class 11 — systematic practice reliably produces high scores.
Limits: The 3 Methods
1. Direct Substitution
If substituting x = a gives a definite value (not 0/0), that IS the limit.
2. Factorisation Method
When direct substitution gives 0/0: factorise, cancel the common factor, then substitute.
3. Rationalisation Method
When the expression contains a square root: multiply by the conjugate.
Standard Limits to Memorise
- lim(x to 0) sin(x)/x = 1
- lim(x to 0) tan(x)/x = 1
- lim(x to 0) (1 - cos x)/x = 0
- lim(x to a) (x^n - a^n)/(x - a) = n x a^(n-1)
Derivatives: The 4 Rules
| Rule | Formula | Example |
|---|---|---|
| Power Rule | d/dx(x^n) = n x x^(n-1) | d/dx(x^5) = 5x^4 |
| Product Rule | d/dx(uv) = u.v' + v.u' | d/dx(x.sin x) = sin x + x.cos x |
| Quotient Rule | d/dx(u/v) = (v.u' - u.v')/v^2 | d/dx(sin x/x) |
| Chain Rule | d/dx f(g(x)) = f'(g(x)).g'(x) | d/dx(sin(x^2)) = 2x.cos(x^2) |
Tip
Derivatives from First Principles appear every year for 3 marks. Formula: f'(x) = lim(h to 0) [f(x+h) - f(x)]/h. Practise this for sin x, cos x, x^n, and constant — these four cover all board-level questions.