Class 11 Maths NCERT Solutions Chapter 12 Limits and Derivatives | Boundless Maths
Chapter 12 · Class 11 Maths NCERT Solutions

Class 11 Maths NCERT Solutions Chapter 12 — Limits and Derivatives

Every question from Exercise 12.1, Exercise 12.2 and the Miscellaneous Exercise, solved step by step — the introduction to calculus, built on limits and the first principle of a derivative.

Limits and Derivatives is Class 11's first real brush with calculus. The chapter opens with the intuitive idea of a limit — what a function's value should be near a point, even if it isn't defined there — then formalises this through left-hand and right-hand limits and the algebra of limits. From there it defines the derivative, first from the raw limit definition (the first principle), then through the sum, product and quotient rules that make differentiating everyday functions fast.

3Exercises (incl. Misc.)
73Total Questions
2026-27CBSE Syllabus
100%Solved

Class 11 Maths NCERT Solutions Chapter 12 — Overview

This Class 11 Maths NCERT Solutions Chapter 12 hub covers Limits and Derivatives — the chapter that formally opens calculus. A limit asks what value a function should take at a point, based only on its values nearby, without ever requiring the function to actually be defined at that point. This idea is split into a left-hand limit (approaching from below) and a right-hand limit (approaching from above); the limit exists only when the two agree.

Once limits are established, the chapter builds two standard trigonometric results — \lim_{x\to0}\frac{\sin x}{x}=1 and \lim_{x\to0}\frac{1-\cos x}{x}=0 — that unlock every trigonometric limit in Exercise 12.1. The second half of the chapter defines the derivative directly as a limit (the first principle), then develops the algebra of derivatives — the sum, difference, product and quotient rules — so that later chapters can differentiate without going back to first principles every time.

Exercise 12.1 (32 questions) is entirely about evaluating limits. Exercise 12.2 (11 questions) introduces derivatives, insisting on the first principle for a few questions before allowing the algebra of derivatives for the rest. The Miscellaneous Exercise (30 questions) is mixed practice, mostly derivatives of algebraic and trigonometric functions with literal constants standing in for numbers.

How The Chapter Builds

From the Idea of a Limit to the Algebra of Derivatives

1

Intuitive Idea of a Limit

What value should f(x) take near x = a? Left-hand and right-hand limits, and when the limit exists.

→
2

Algebra of Limits

Evaluating limits of polynomial and rational functions by direct substitution, factoring, and cancelling. Exercise 12.1.

→
3

Standard Trigonometric Limits

sin(x)/x → 1 and (1 − cos x)/x → 0 as x → 0, used to solve every trig limit in the exercise. Exercise 12.1.

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4

First Principle of a Derivative

f′(x) defined directly as a limit of the difference quotient — the definition every shortcut rule is built from. Exercise 12.2.

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5

Algebra of Derivatives

Sum, product and quotient rules, plus standard formulas for xⁿ, sin x and cos x. Exercise 12.2 and the Miscellaneous Exercise.

Quick Reference

Important Formulas — Chapter 12

Everything you need before you start solving. This is a summary for quick recall — the Formula Cards above have the full printable version.

Limits — Definitions and Algebra

Left-hand and right-hand limits

\lim_{x\to a^-}f(x)\ \text{and}\ \lim_{x\to a^+}f(x)

The limit at x = a exists, and equals this common value, only when both one-sided limits exist and agree.

Algebra of limits

\lim[f\pm g]=\lim f\pm\lim g,\quad \lim[fg]=\lim f\cdot\lim g,\quad \lim\frac{f}{g}=\frac{\lim f}{\lim g}

Valid whenever the individual limits exist (and, for the quotient, the denominator's limit is non-zero).

Standard polynomial limit

\lim_{x\to a}\frac{x^n-a^n}{x-a}=na^{n-1}

The key tool for 0/0 forms in rational functions — also valid for rational powers n.

Standard Trigonometric Limits

Sine over x

\lim_{x\to0}\frac{\sin x}{x}=1

Used after rewriting any sin(kx)/x-type expression to match this exact form.

One minus cosine over x

\lim_{x\to0}\frac{1-\cos x}{x}=0

Comes from the identity 1 − cos x = 2sin²(x/2).

Derivatives — First Principle and Algebra

First principle

f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}

The definition of the derivative — every standard formula below is derived from this.

Power rule

\frac{d}{dx}(x^n)=nx^{n-1}

Valid for any real number n.

Standard trig derivatives

\frac{d}{dx}(\sin x)=\cos x,\quad \frac{d}{dx}(\cos x)=-\sin x,\quad \frac{d}{dx}(\tan x)=\sec^2x

Also secx tanx for secx, and −cosecx cotx for cosecx.

Product rule

(uv)'=u'v+uv'

Used whenever a function is a product of two simpler functions.

Quotient rule

\left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2}

Used whenever a function is one expression divided by another (v ≠ 0).

Method Selection

Which Technique Should You Use?

If you see...Use this approach
Direct substitution gives a real, defined valueThe limit equals the value of the function there — no further work needed.
Substitution gives 0/0 with a polynomial or rational functionFactor numerator and denominator, cancel the common factor, then substitute again.
Substitution gives 0/0 with sin, cos, or tan expressionsRewrite to match sin(x)/x → 1 or (1 − cos x)/x → 0, using angle identities if needed.
A function is defined piecewise, at the breakpointCompute the left-hand limit and right-hand limit separately using the matching piece of the definition; the limit exists only if they agree.
The question says "from first principle"Use f′(x) = lim(h→0) [f(x+h) − f(x)]/h directly — do not use shortcut differentiation rules.
A function is a product or quotient of simpler functionsApply the product rule or quotient rule, using the standard derivatives of each piece.
Avoid These

Common Mistakes to Avoid in This Chapter

Drawn from where students actually lose marks across all three exercises.

  • Assuming the limit equals the function's value at that point without checking. This only holds when the function is actually defined and continuous there — for a 0/0 form, substitution tells you nothing until you factor and cancel.
  • Forgetting to check both one-sided limits for a piecewise function. A limit "existing" always means the left-hand and right-hand limits agree — checking only one side is an incomplete answer.
  • Misapplying the standard trig limits without rewriting first. lim sin(ax)/x is not simply 1 — it must be rewritten as a × [sin(ax)/(ax)] before the standard result applies.
  • Using differentiation shortcuts on a "from first principle" question. Examiners specifically check for the limit-definition working; jumping straight to a known derivative formula loses method marks even if the final answer is correct.
  • Sign errors differentiating cos x, cot x and cosec x. All three carry a negative sign in their derivative — cos x → −sin x, cot x → −cosec²x, cosec x → −cosecx cotx — and it's easy to drop the sign under time pressure.
  • Not simplifying before differentiating. Expanding a product like x⁻³(5 + 3x) first, or cancelling a common factor in (xⁿ − aⁿ)/(x − a), is often far faster than applying the product or quotient rule directly.
Solve Chapter-Wise

Class 11 Maths NCERT Solutions Chapter 12 — Choose an Exercise

12.1

Exercise 12.1

Limits — algebraic and rational limits by substitution and factoring, standard trigonometric limits, and left-hand/right-hand limits of piecewise functions · 32 questions

Solve Exercise 12.1 →
12.2

Exercise 12.2

Derivatives — from the first principle, and using the sum, product and quotient rules on polynomial and trigonometric functions · 11 questions

Solve Exercise 12.2 →
M

Miscellaneous Exercise

Mixed derivative practice — first principle once more, then a wide range of algebraic and trigonometric derivatives with literal constants · 30 questions

Solve Miscellaneous →

Revising before a test?

Every definition and result from this chapter — the algebra of limits, standard trigonometric limits, the algebra of derivatives, standard derivative formulas — on one printable formula sheet.

Get Formula Cards →
Common Questions

Class 11 Maths NCERT Solutions Chapter 12 — FAQs

How many exercises and questions are there in Chapter 12?
Chapter 12, Limits and Derivatives, has two numbered exercises plus a Miscellaneous Exercise. Exercise 12.1 (Limits) has 32 questions, Exercise 12.2 (Derivatives) has 11 questions, and the Miscellaneous Exercise has 30 questions — 73 questions in total.
What is the difference between a limit and a derivative?
A limit describes the value a function approaches as its input gets closer to some point, without necessarily reaching it — it is the foundational idea behind all of calculus. A derivative is built directly from a limit: it is defined as f′(x) = lim(h→0) [f(x+h) − f(x)] / h, and it measures the instantaneous rate of change of a function at a point, or equivalently the slope of the tangent to its graph there.
What are the two standard trigonometric limits used throughout this chapter?
The two results used repeatedly are lim(x→0) sin(x)/x = 1 and lim(x→0) [1 − cos(x)]/x = 0. Most trigonometric limit questions in Exercise 12.1 are solved by algebraically rewriting the given expression so that one or both of these standard forms appear, then applying the result directly.
What is the first principle of finding a derivative?
The first principle (or the limit definition) of the derivative says that for a function f, its derivative at x is f′(x) = lim(h→0) [f(x+h) − f(x)] / h, provided this limit exists. Several questions in Exercise 12.2 and the Miscellaneous Exercise require this definition directly, without using shortcut differentiation rules, since it is how those very rules are proved in the first place.
Where can I find the official NCERT textbook for this chapter?
Limits and Derivatives is Chapter 12 of the NCERT Class 11 Mathematics textbook, published by the National Council of Educational Research and Training (NCERT) and prescribed by CBSE. You can download the official textbook PDF directly from ncert.nic.in, NCERT's official website — the solutions on this page follow the exercises exactly as they appear there.
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