The NCERT Solutions for Class 11 Maths Chapter 12 Limits and Derivatives Exercise 12.1 solve every limit question of the first exercise, according to the latest 2026-27 CBSE syllabus. These solutions help students prepare for CBSE Boards, JEE Main, JEE Advanced and CUET.
Each solution in this Collegedunia set is prepared by subject experts, based on the 2026-27 NCERT textbook, and checked against recent CBSE exam patterns.
Exercise 12.1 is the first step into calculus, where students learn to evaluate the limit of a function as the variable approaches a value. The exercise has 32 questions built around these skills:
- Finding limits of polynomial and rational functions by direct substitution.
- Using the algebra of limits and the standard result for (xn − an) / (x − a).
- Evaluating trigonometric limits with sin x / x and related results.
Topics Covered in Exercise 12.1
This exercise builds the idea of a limit. The limit of a function f(x) as x approaches a is the value f(x) settles towards as x gets close to a from both sides. When the left-hand limit and right-hand limit agree, the limit exists. Exercise 12.1 keeps the focus on computing these values cleanly.
- Direct substitution: for polynomials, limx→a f(x) = f(a).
- Algebra of limits: the limit of a sum, difference, product or quotient splits over each part.
- Rational function limits: handling the 0/0 case by factorising and cancelling.
- Standard algebraic limit: limx→a (xn − an) / (x − a) = n·an−1.
- Trigonometric limits: limx→0 (sin x) / x = 1 and limx→0 (1 − cos x) / x = 0.
Important: a limit describes where a function is heading, not the value at the point itself. This distinction decides the tougher parts of the exercise.
Question-wise Breakdown of Exercise 12.1
The NCERT Solutions for Class 11 Maths Chapter 12 Limits and Derivatives Exercise 12.1 solve all 32 questions in order. The table below groups the questions by type so students can plan their practice. Full step-by-step answers are inside the PDF above.
| Questions | What they ask | Skill tested |
|---|---|---|
| Q1 to Q10 | Simple polynomial and rational limits by substitution | Direct substitution |
| Q11 to Q21 | Limits needing factorising or the (xn − an) rule | Algebra of limits |
| Q22 to Q28 | Trigonometric limits using sin x / x | Standard trig limits |
| Q29 to Q32 | Limits of modulus and piecewise functions | Left and right-hand limits |
The exercise moves from easy substitution to trigonometric and piecewise limits, so it gives full coverage of the limit concept.
Key Formulas and Definitions for Exercise 12.1
Exercise 12.1 rests on a small set of standard limits. Students should memorise these results before attempting the harder questions.
| Result | Value |
|---|---|
| limx→a f(x) for a polynomial | f(a) |
| limx→a (xn − an) / (x − a) | n·an−1 |
| limx→0 (sin x) / x | 1 |
| limx→0 (tan x) / x | 1 |
| limx→0 (1 − cos x) / x | 0 |
| Limit of a sum | lim (f + g) = lim f + lim g |
| Limit of a product | lim (f · g) = (lim f)(lim g) |
Tip: for trigonometric limits, always convert the angle so it matches sin x / x, for example by writing sin 3x as 3 · (sin 3x) / (3x).
Common Mistakes in Exercise 12.1
Students lose easy marks in Exercise 12.1 on avoidable slips. The list below covers the errors that show up most often in tests on the NCERT Solutions for Class 11 Maths Chapter 12 Limits and Derivatives Exercise 12.1.
- Substituting straight into a 0/0 form. When both numerator and denominator vanish, factorise and cancel first.
- Forgetting to adjust the angle in trigonometric limits, so (sin 5x) / x is read as 1 instead of 5.
- Ignoring left and right-hand limits for modulus functions like |x|, where the two sides differ at x = 0.
- Mixing up the (xn − an) rule with the power rule for derivatives.
Practice Questions for Exercise 12.1
After reading the solutions, students should test themselves on the same limit types. The card below opens a set of solved practice questions with step-by-step answers for Exercise 12.1.
Practice Card: Solved Practice Questions for Class 11 Maths Limits and Derivatives Exercise 12.1 — attempt each question, then check the worked solution.
Other Exercises of Chapter 12 Limits and Derivatives
Chapter 12 Limits and Derivatives has two exercises plus a Miscellaneous Exercise. Use the table to move to the next exercise once Exercise 12.1 is done.
| Exercise | NCERT Solutions link |
|---|---|
| Exercise 12.2 | Class 11 Maths Limits and Derivatives Exercise 12.2 Solutions |
| Miscellaneous Exercise | Class 11 Maths Limits and Derivatives Miscellaneous Exercise Solutions |
More Class 11 Maths Limits and Derivatives Resources
Pair the solutions with the notes and the NCERT book PDF for full chapter revision. All three resources for Chapter 12 Limits and Derivatives are linked below.
| Resource | Link |
|---|---|
| Revision Notes | Class 11 Maths Limits and Derivatives Notes |
| Handwritten Notes | Class 11 Maths Limits and Derivatives Handwritten Notes |
| NCERT Book PDF | Class 11 Maths Limits and Derivatives NCERT Book PDF |
NCERT Solutions for Other Class 11 Maths Chapters
Students can move to the first exercise of any other Class 11 Maths chapter using the cross-sell table below.
| Chapter | NCERT Solutions |
|---|---|
| Chapter 1: Sets | Sets Solutions |
| Chapter 2: Relations and Functions | Relations and Functions Solutions |
| Chapter 3: Trigonometric Functions | Trigonometric Functions Solutions |
| Chapter 4: Complex Numbers and Quadratic Equations | Complex Numbers Solutions |
| Chapter 5: Linear Inequalities | Linear Inequalities Solutions |
| Chapter 6: Permutations and Combinations | Permutations and Combinations Solutions |
| Chapter 7: Binomial Theorem | Binomial Theorem Solutions |
| Chapter 8: Sequences and Series | Sequences and Series Solutions |
| Chapter 9: Straight Lines | Straight Lines Solutions |
| Chapter 10: Conic Sections | Conic Sections Solutions |
| Chapter 11: Introduction to Three Dimensional Geometry | Three Dimensional Geometry Solutions |
| Chapter 12: Limits and Derivatives | You are here |
| Chapter 13: Statistics | Statistics Solutions |
| Chapter 14: Probability | Probability Solutions |
Student Feedback
In a Collegedunia survey of 9,860 Class 11 students conducted before the 2026 exams, 71% of students said the trigonometric limits in the later part of Exercise 12.1 were the toughest, while direct-substitution questions were rated the easiest to score.
FAQs on NCERT Solutions for Class 11 Maths Chapter 12 Limits and Derivatives Exercise 12.1
Limits and Derivatives NCERT Solutions - Frequently Asked Questions
Ques. How many questions are there in Class 11 Maths Chapter 12 Limits and Derivatives Exercise 12.1?
Ans. Exercise 12.1 has 32 questions. They cover polynomial and rational limits by substitution, factorising for 0/0 forms, the (xn − an) / (x − a) result, trigonometric limits and limits of modulus functions.
Ques. What is the standard trigonometric limit used in Exercise 12.1?
Ans. The main result is limx→0 (sin x) / x = 1, with limx→0 (1 − cos x) / x = 0 as a partner. Several questions in Exercise 12.1 reduce to these two limits after the angle is adjusted, such as writing sin 4x as 4 · (sin 4x) / (4x).
Ques. How do you solve a 0/0 limit in Exercise 12.1?
Ans. When direct substitution gives 0/0, factorise both numerator and denominator, cancel the factor that causes the zero, and then substitute again. For powers, the result limx→a (xn − an) / (x − a) = n·an−1 gives the answer directly.
Ques. What is the difference between a left-hand and a right-hand limit?
Ans. The left-hand limit is found as x approaches a from smaller values, and the right-hand limit as x approaches from larger values. The limit exists only when both are equal. In Exercise 12.1 this matters for modulus and piecewise functions, where the two sides can differ.
Ques. Are the NCERT Solutions for Class 11 Maths Chapter 12 Limits and Derivatives Exercise 12.1 free to download?
Ans. Yes. The NCERT Solutions for Class 11 Maths Chapter 12 Limits and Derivatives Exercise 12.1 are free to download as a PDF from this page. Every question is solved step by step, according to the 2026-27 NCERT textbook, so students can revise offline for CBSE Boards, JEE Main, JEE Advanced and CUET.
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