∞ CBSE · Class 11 · Mathematics · Chapter 13

Limits and
Derivatives

Complete chapter resources for CBSE Class 11 Mathematics — topic breakdown, key formulas, sample questions, previous year board questions, and instant AI question paper generation.

3Topics
6–8Board marks
8Sample questions
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Key Formulas — Chapter 13
  • Standard limit: lim(x→a) (xⁿ − aⁿ)/(x − a) = naⁿ⁻¹
  • Trig limit: lim(x→0) sin x / x = 1
  • Trig limit: lim(x→0) (1 − cos x) / x = 0
  • Derivative (1st principles): f'(x) = lim(h→0) [f(x+h) − f(x)] / h
  • Power rule: d/dx (xⁿ) = nxⁿ⁻¹
  • Product rule: (uv)' = u'v + uv'
Chapter Overview

What this chapter covers

Chapter 13 introduces the foundational ideas of calculus at the Class 11 level. The concept of a limit formalises the intuitive idea of a function's value "approaching" a specific number as the input gets arbitrarily close to a given point — without necessarily reaching it. The chapter begins with an intuitive treatment of limits using graphs and tables, then progresses to algebraic evaluation techniques such as direct substitution, factorisation, rationalisation, and applying standard results.

The second half of the chapter defines the derivative as the limit of the difference quotient: f'(x) = limh→0 [f(x+h) − f(x)] / h. This first-principles approach is the rigorous foundation of differentiation. Students learn to derive standard derivatives — polynomials, trigonometric functions — from scratch using this definition, and then apply differentiation rules (sum, product, quotient) to combine results efficiently.

Board questions from this chapter test three distinct skills: evaluating algebraic and trigonometric limits, computing derivatives using first principles, and applying differentiation rules to composite expressions. Word-problem style questions are uncommon — the chapter is primarily computational, making formula fluency and step presentation the most critical exam skills.

Topics

What's inside Chapter 13

As per NCERT Class 11 Mathematics (CBSE syllabus)

Topic 1
Limits of Functions
Intuitive definition of a limit. Left-hand and right-hand limits. Existence condition: LHL = RHL. Algebra of limits (sum, difference, product, quotient rules). Standard limits including lim(x→a)(xⁿ − aⁿ)/(x − a) = naⁿ⁻¹ and key trigonometric limits at x → 0.
Topic 2
Derivatives — First Principles
Definition of the derivative as the limit of the difference quotient. Geometric interpretation as the slope of a tangent. Derivation of derivatives of xⁿ, sin x, and cos x from first principles. Conditions for differentiability.
Topic 3
Algebra of Derivatives
Sum and difference rules: (u ± v)' = u' ± v'. Product rule: (uv)' = u'v + uv'. Quotient rule: (u/v)' = (u'v − uv') / v². Derivatives of standard trigonometric functions — sin x, cos x, tan x, cot x, sec x, cosec x.
Connections

How this chapter fits in

Useful for setting question difficulty and cross-chapter papers.

Builds on
Ch 3 · Trigonometric Functions
Trig identities and values needed for limit evaluation and derivatives of sin x / cos x
Ch 2 · Relations and Functions
Function notation, domain-range concepts, and polynomial function behaviour
Chapter 13 Limits &
Derivatives
Leads to
Class 12 · Continuity & Differentiability
Chain rule, implicit differentiation, Rolle's and Mean Value Theorems
Class 12 · Applications of Derivatives
Tangents, normals, maxima and minima, increasing/decreasing functions
Board Blueprint

Marks & question-type breakdown

Typical pattern based on CBSE Class 11 Mathematics papers from the last five years.

Question type Marks Typical count What's usually tested
MCQ / Objective 1 1 Direct limit evaluation or identify a standard derivative
Very Short Answer 2 1 Evaluate an algebraic or trigonometric limit using standard results
Short Answer 3 1 Differentiate using product/quotient rule or evaluate a combined limit
Long Answer — First Principles 4–5 1 Derive the derivative of sin x, cos x, or a polynomial from first principles
Total (approximate) 6–8 4 Weightage varies across paper sets and years
Sample Questions

8 sample questions — generated by MarksZen AI

Aligned to CBSE Class 11 Mathematics Chapter 13. Covers all question types across Easy, Medium, and Hard difficulty.

Q1 Easy 1 mark MCQ
The value of lim(x→2) (x² − 4)/(x − 2) is: (a) 0 (b) 2 (c) 4 (d) undefined
Q2 Easy 2 marks Short Answer
Evaluate: lim(x→0) sin 3x / x
Q3 Medium 2 marks Short Answer
Find the derivative of f(x) = 3x² − 5x + 7 using the power rule. State the value of f'(2).
Q4 Medium 3 marks Short Answer
Evaluate: lim(x→0) (√(1 + x) − 1) / x (Hint: rationalise the numerator.)
Q5 Medium 3 marks Short Answer
Differentiate f(x) = (x² + 1)(2x − 3) using the product rule. Verify by expanding the product first and differentiating directly.
Q6 Hard 4 marks Short Answer
Find the derivative of f(x) = (x² + sin x) / cos x using the quotient rule. Simplify your answer.
Q7 Hard 5 marks Long Answer
Using the first-principles definition of the derivative, prove that d/dx (sin x) = cos x. (Use the identities: sin(A+B) − sin A = 2 cos((2A+B)/2) sin(B/2), and lim(h→0) sin h/h = 1.)
Q8 Hard 5 marks Long Answer
Find the derivative of f(x) = xⁿ from first principles, where n is a positive integer. Hence find the derivative of f(x) = x⁴ − 3x³ + 2x − 1 and evaluate f'(1).
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Previous Year Questions

From CBSE board examinations

Actual questions from past Class 11 Mathematics board papers — Limits and Derivatives chapter.

Board 20222 marks
Evaluate: lim(x→π/2) (tan 2x) / (x − π/2) (All India 2022)
Board 20233 marks
Differentiate f(x) = (x + cos x)(x − tan x) with respect to x. (Delhi 2023)
Board 20204 marks
Find the derivative of cos x from first principles. (CBSE 2020)
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FAQ

Questions teachers ask

How many marks does Limits and Derivatives carry in the CBSE Class 11 Mathematics exam? +
Limits and Derivatives typically carries 6–8 marks in the CBSE Class 11 end-term examination. Questions usually appear as one 2-mark short answer (evaluating a limit), one 3-mark derivative problem, and occasionally a 4-mark case involving first-principles or product/quotient rule. The exact split varies by paper set.
What is the difference between a limit and a derivative in Class 11? +
A limit describes the value a function approaches as the input approaches a point — it does not require the function to be defined at that point. A derivative is defined as the limit of the difference quotient [f(x+h) − f(x)] / h as h → 0, and it measures the instantaneous rate of change of a function at a specific point. Limits are the foundational tool used to define the derivative.
Do CBSE Class 11 board exams ask for proofs using the first-principles definition of a derivative? +
Yes. First-principles (limit of the difference quotient) derivations are explicitly part of the NCERT syllabus and appear as 3–4 mark questions. Common asks include: prove d/dx(xⁿ) = nxⁿ⁻¹ for positive integers, or derive the derivative of sin x or cos x from first principles. Students must know the step-by-step expansion and limit evaluation.
Which standard limit formulas must a Class 11 student memorise for board exams? +
The most tested standard limits are: lim(x→a) (xⁿ − aⁿ)/(x − a) = naⁿ⁻¹; lim(x→0) sin x / x = 1; lim(x→0) (1 − cos x) / x = 0; lim(x→0) tan x / x = 1. These appear in direct substitution, factorisation, and trigonometric limit questions. The first is the algebraic standard limit and the sin x / x result is the most frequently tested trigonometric limit.
How do I generate a custom question paper for Limits and Derivatives using MarksZen? +
Sign up for a free MarksZen account, choose CBSE Class 11 Mathematics, select Chapter 13 (Limits and Derivatives), set your preferred question-type mix (MCQ, short answer, first-principles proof) and total marks — the AI generates a complete board-aligned paper with answer key in under 2 minutes, ready for PDF export.