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.
Standard Limits — Exponential and Logarithmic
The CBSE 2025-26 syllabus explicitly includes limits of exponential and logarithmic functions. Two standard results you must memorise:
- lim(x to 0) (e^x - 1)/x = 1
- lim(x to 0) log(1 + x)/x = 1 (natural log)
Derivatives of Trigonometric Functions
The CBSE syllabus requires derivatives of polynomial and trigonometric functions. Memorise these six results — they appear directly as 1-mark questions or as steps inside larger problems:
| Function | Derivative |
|---|---|
| sin x | cos x |
| cos x | −sin x |
| tan x | sec² x |
| cosec x | −cosec x · cot x |
| sec x | sec x · tan x |
| cot x | −cosec² x |
How Calculus Is Examined in Class 11
Calculus (Unit IV) carries 8 marks in the Class 11 annual examination out of 80. This means one or two questions — typically one 3-mark and one 5-mark, or two 4-mark questions. The chapter is not a high-mark unit, which is exactly why students who prepare it properly can score full marks here without difficulty.
- 3-mark question: evaluate a limit using one of the 3 methods, or state and verify a standard limit result
- 5-mark question: differentiate from First Principles for a given function (sin x, cos x, x^n are the most common)
- 1-mark questions: direct derivatives of polynomial or trig functions using rules
Common Mistakes That Cost Marks
- Applying L'Hôpital's Rule — this is NOT in the CBSE Class 11 syllabus. Use the three methods only.
- Skipping the 'lim' notation while writing limit steps — the examiner expects 'lim(x→a)' on every line until the final substitution
- Confusing Product Rule with Chain Rule: product rule is for two multiplied functions of x; chain rule is for a function of a function
- Writing d/dx(sin x) = −cos x — the derivative of sin x is +cos x. The negative sign belongs to cos x
- In First Principles, expanding f(x+h) incorrectly — write out the expansion fully and then simplify, never skip steps
The Right Practice Sequence
- 1.Week 1: Master the 3 limit methods — 5 problems each on direct substitution, factorisation, and rationalisation
- 2.Week 2: Memorise all standard limits — write them from memory daily. Then evaluate 10 problems using them
- 3.Week 3: Learn the derivative rules in order — Power, Product, Quotient. Do 5 problems each
- 4.Week 4: Memorise all 6 trig derivatives. Practise First Principles for x^n, sin x, and cos x until you can write them from memory
- 5.Final pass: Solve 5 previous year Class 11 annual exam papers. All Calculus questions will be covered
Tip
Calculus is 8 out of 80 marks. Students who practise First Principles for one week can reliably score all 5 marks from it. That single investment returns more than equivalent time spent on any other 8-mark unit in Class 11 Maths.