Limits and Derivatives – Class 11 Maths Chapter 12 Notes & NCERT PDF

Class 11MathematicsChapter 12NCERT book: Mathematics - Class 11

Notes and a simple summary of Chapter 12 of the NCERT Class 11 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 12: Limits and Derivatives

Introduction

Chapter 12, Limits and Derivatives, opens the door to calculus — the branch of mathematics that studies how quantities change. The book starts with an intuitive idea of derivatives: a body falling freely covers a distance s = 4.9t² metres in t seconds, and by computing average velocities over smaller and smaller time intervals around t = 2, we "close in" on the instantaneous velocity at t = 2. This process of approaching a value is exactly what a limit captures, and the instantaneous rate of change is the derivative.

Key Concepts and Theorems

12.3 Limits

lim f(x) as x → a is the value f(x) approaches as x gets closer and closer to a (not necessarily f(a) itself).

Left hand limit (LHL): the value f(x) approaches when x comes to a from the left (x < a). Right hand limit (RHL): from the right (x > a).

The limit exists only when LHL = RHL, and then lim f(x) equals that common value.

f(a) and lim f(x) as x → a may be different — one may even exist when the other does not.

"Limit of a function at a point is the common value of the left and right hand limits, if they coincide."

12.3.1 Algebra of limits (Theorem 1)

If lim f(x) and lim g(x) both exist as x → a, then:

lim [f(x) ± g(x)] = lim f(x) ± lim g(x)

lim [f(x)·g(x)] = lim f(x) · lim g(x)

lim [f(x)/g(x)] = lim f(x) / lim g(x), provided lim g(x) ≠ 0

lim [λ·f(x)] = λ·lim f(x) for any constant λ

12.3.2 Limits of polynomials and rational functions

For a polynomial, lim f(x) as x → a is simply f(a) — substitute directly.

For a rational function p(x)/q(x): substitute first. If you get the form 0/0, factorise and cancel the common factor (x − a), then substitute again.

Theorem 2: for any positive integer n, lim (xⁿ − aⁿ)/(x − a) as x → a = n·aⁿ⁻¹. (The book notes it holds for any rational n and positive a.)

12.4 Limits of trigonometric functions

Theorem 3: if f(x) ≤ g(x) near a and both limits exist, then lim f ≤ lim g.

Theorem 4 (Sandwich Theorem): if f(x) ≤ g(x) ≤ h(x) and lim f(x) = lim h(x) = l, then lim g(x) = l.

Two standard limits (proved using the sandwich theorem with areas of a sector and triangles):

- lim (sin x)/x as x → 0 = 1

- lim (1 − cos x)/x as x → 0 = 0

12.5 Derivatives

Definition: the derivative of f at a is f′(a) = lim [f(a + h) − f(a)]/h as h → 0, if this limit exists.

The derivative function: f′(x) = lim [f(x + h) − f(x)]/h. Finding it straight from this definition is called differentiating from first principles. Notations: f′(x), d/dx f(x), dy/dx, y′.

12.5.1 Algebra of derivatives (Theorem 5)

(u ± v)′ = u′ ± v′

Product rule (Leibnitz rule): (uv)′ = u′v + uv′

Quotient rule: (u/v)′ = (u′v − uv′)/v², v ≠ 0

Theorem 6: d/dx(xⁿ) = n·xⁿ⁻¹ for any positive integer n.

Theorem 7: a polynomial aₙxⁿ + … + a₁x + a₀ has derivative naₙxⁿ⁻¹ + … + a₁.

Worked Examples

Polynomial limits (Example 1): lim (x³ − x² + 1) as x → 1 = 1 − 1 + 1 = 1; lim x(x + 1) as x → 3 = 3 × 4 = 12; lim (1 + x + x² + … + x¹⁰) as x → −1 = 1 − 1 + 1 − … + 1 = 1.

Using Theorem 2 (Example 3): lim (x¹⁵ − 1)/(x¹⁰ − 1) as x → 1 = [lim (x¹⁵ − 1)/(x − 1)] ÷ [lim (x¹⁰ − 1)/(x − 1)] = 15 ÷ 10 = 3/2.

Trigonometric limits (Example 4): lim sin 4x / sin 2x as x → 0 = 2 · [4·(sin 4x/4x)] ÷ [2·(sin 2x/2x)] ÷ 2… simplifying gives 2; and lim tan x / x as x → 0 = lim (sin x/x)·(1/cos x) = 1 × 1 = 1.

Derivative at a point (Example 6): f(x) = 2x² + 3x − 5 ⇒ f′(x) = 4x + 3, so f′(−1) = −1 and f′(0) = 3, hence f′(0) + 3f′(−1) = 3 − 3 = 0, as the book asks us to prove.

Polynomial rule (Example 13): d/dx(6x¹⁰⁰ − x⁵⁵ + x) = 600x⁹⁹ − 55x⁵⁴ + 1.

Clever use of Theorem 6 (Example 14): f(x) = 1 + x + x² + … + x⁵⁰ ⇒ f′(x) = 1 + 2x + … + 50x⁴⁹, so f′(1) = 1 + 2 + … + 50 = 1275.

Trig derivatives (Examples 16–18): (sin x)′ = cos x; (tan x)′ = sec²x (quotient rule on sin x / cos x); (sin²x)′ = 2 sin x cos x = sin 2x (product rule).

Formula Sheet

lim (xⁿ − aⁿ)/(x − a), x → a = n·aⁿ⁻¹

lim (sin x)/x, x → 0 = 1; lim (1 − cos x)/x, x → 0 = 0

f′(x) = lim [f(x + h) − f(x)]/h, h → 0

d/dx(xⁿ) = nxⁿ⁻¹; d/dx(constant) = 0

d/dx(sin x) = cos x; d/dx(cos x) = −sin x; d/dx(tan x) = sec²x

(uv)′ = u′v + uv′; (u/v)′ = (u′v − uv′)/v²

Exam Focus

Evaluate limits (Exercise 12.1): always substitute first; if 0/0, factorise, rationalise, or use a standard limit.

- Outline: lim (x² − 9)/(x − 3), x → 3 → factor (x − 3)(x + 3)/(x − 3) → x + 3 → 6.

First principles (Exercise 12.2): write f(x + h) − f(x), divide by h, simplify, then let h → 0.

- Outline for f(x) = x²: [(x + h)² − x²]/h = 2x + h → 2x.

Product/quotient rules: state the rule, identify u and v, substitute carefully.

- Outline: d/dx(x sin x) = 1·sin x + x·cos x = sin x + x cos x; d/dx[x/(1 + x)] = [(1 + x) − x]/(1 + x)² = 1/(1 + x)².

Key Terms

Limit (सीमा) — the value a function approaches

Left hand limit / Right hand limit (वाम / दक्षिण पक्ष सीमा)

Derivative (अवकलज) — instantaneous rate of change

First principle (प्रथम सिद्धांत) — finding a derivative from the definition

Instantaneous velocity (तात्कालिक वेग)

Polynomial function (बहुपद फलन)

Rational function (परिमेय फलन)

Indeterminate form 0/0 (अनिर्धार्य रूप)

Sandwich theorem (सैंडविच प्रमेय)

Product rule / Quotient rule (गुणन / भागफल नियम)

Differentiation (अवकलन)

Tangent / slope (स्पर्श रेखा / ढाल)

Common Mistakes

Assuming lim f(x) must equal f(a) — the function may not even be defined at a.

Stopping at 0/0 and writing "undefined": 0/0 means more work is needed, not that the limit fails.

Using degrees in lim (sin x)/x = 1 — the result holds only when x is in radians.

Writing (uv)′ = u′v′ or (u/v)′ = u′/v′ — always use the full product and quotient rules.

Forgetting the minus sign: d/dx(cos x) = −sin x.

In first principles, dividing by h before expanding and cancelling properly.

💡 Key Learning Points

✓A limit exists at x = a only when the left hand limit and right hand limit are equal; it need not equal f(a).
✓Polynomial limits are found by direct substitution; 0/0 forms of rational functions are handled by factorising and cancelling.
✓Standard limits: lim (xⁿ − aⁿ)/(x − a) = n·aⁿ⁻¹, lim (sin x)/x = 1 and lim (1 − cos x)/x = 0 as x → 0.
✓The derivative f′(a) = lim [f(a + h) − f(a)]/h measures the instantaneous rate of change, e.g. instantaneous velocity.
✓Derivatives follow the sum, product (Leibnitz) and quotient rules, with d/dx(xⁿ) = nxⁿ⁻¹, (sin x)′ = cos x, (cos x)′ = −sin x.

👨‍🏫 Teaching Tips

→Recreate the book's falling-body table (s = 4.9t²) on the board and let students compute average velocities over shrinking intervals around t = 2 to feel what 'approaching' means.
→Use a two-column LHL/RHL table for piecewise functions before introducing the formal statement that the limit is their common value.
→Drill a fixed routine for limits: substitute → check for 0/0 → factorise/rationalise/standard limit → substitute again.
→Show the sandwich-theorem proof of lim (sin x)/x = 1 with the unit-circle diagram, and stress that x must be in radians.
→Have students derive d/dx(x²) and d/dx(x³) from first principles before revealing the general xⁿ rule.
→Practise product and quotient rules on (sin²x)′ and (tan x)′ exactly as in Examples 17–18, then ask students to verify with the chain of simpler rules.
→End each class with a mixed 5-question 'limit or derivative?' sprint to build fluency.

📋 Assessment Questions

Can the student evaluate limits of polynomial and rational functions, including 0/0 forms?
Can the student apply lim (xⁿ − aⁿ)/(x − a) and the two standard trigonometric limits correctly?
Can the student find a derivative from first principles for simple functions like x², 1/x or sin x?
Can the student apply the product and quotient rules without sign or ordering errors?
Can the student interpret the derivative as a rate of change (e.g. instantaneous velocity)?

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