NCERT Notes Class 11 Maths Chapter 12 Limits and Derivatives

The class 11 maths notes chapter 12 limits and derivatives bring together every limit rule, standard result, and derivative formula that the CBSE Boards, JEE Main, JEE Advanced, and CUET actually test in 2026-27. These revision notes explain the intuitive idea of a limit, left and right hand limits, the algebra of limits, standard trigonometric limits, and the derivative from first principle in plain language, with a full formula table for quick recall.

You can download the complete Limits and Derivatives notes PDF above, then use the topic-by-topic summary and the formula table on this page for a fast last-minute revision.

• CBSE Boards: Limits and Derivatives closes the Calculus unit and carries roughly 6 to 8 marks in the Class 11 final paper.
• JEE Main and CUET: expect 1 to 2 direct questions on standard limits, the sin x / x result, and first-principle derivatives every year.
• What is covered: intuitive limits, one-sided limits, algebra of limits, polynomial and rational limits, trigonometric limits, and derivatives.

Every entry in these Limits and Derivatives notes is curated by Collegedunia subject experts, according to the 2026-27 NCERT textbook, and checked against the last five years of CBSE Board and JEE Main papers.

Also Check:

• NCERT Solutions for Class 11 Maths Limits and Derivatives
• Class 11 Maths Limits and Derivatives Handwritten Notes

Topic-by-Topic Summary of Limits and Derivatives for Class 11 Maths

The Limits and Derivatives chapter splits into five sub-topic blocks. The class 11 maths notes chapter 12 limits and derivatives map each block to what the exam asks, so students know where the marks sit before they revise.

• Intuitive idea of limits: the value a function approaches near a point. Usually a 1 to 2-mark concept question.
• Left and right hand limits: approaching a point from below and from above. A common reasoning question on when a limit exists.
• Algebra of limits and polynomials: sum, difference, product, and quotient rules for limits, plus rational-function limits. The 2 to 3-mark core.
• Standard limits: trigonometric limits and the sin x / x result. A frequent 2-mark evaluation question.
• Derivatives: derivative from first principle, the algebra of derivatives, and formulas for powers and trig functions. The 3 to 4-mark heart of the chapter.

The rest of these notes take each block in order, so a first read from top to bottom mirrors the NCERT chapter flow.

Intuitive Idea of Limits and One-Sided Limits

A limit tells you the value a function f(x) gets close to as x moves close to a fixed number a. You do not put x = a; you check what happens just around a. The short way to write this is lim (x → a) f(x).

Sometimes x can approach a from two sides, and the notes handle each side on its own:

• Left hand limit: x approaches a from values smaller than a. Written lim (x → a⁻) f(x).
• Right hand limit: x approaches a from values larger than a. Written lim (x → a⁺) f(x).

The limit exists only when the left hand limit equals the right hand limit. If the two sides give different values, then lim (x → a) f(x) does not exist. This single rule decides most one-mark limit questions in the board paper.

Algebra of Limits, Polynomials and Rational Functions

Once a limit exists, you can break a hard function into easy pieces. The algebra of limits lets you take the limit of each piece and then combine. These rules turn most limit sums into simple substitution.

• Sum and difference: lim [f(x) ± g(x)] = lim f(x) ± lim g(x).
• Product: lim [f(x) · g(x)] = lim f(x) · lim g(x).
• Quotient: lim [f(x) / g(x)] = lim f(x) / lim g(x), provided the bottom limit is not 0.
• Constant multiple: lim [k · f(x)] = k · lim f(x) for any constant k.

For a polynomial, the limit is just direct substitution: lim (x → a) p(x) = p(a). For a rational function p(x)/q(x), substitute first. If it gives the 0/0 form, factor the top and bottom, cancel the common factor, then substitute again. This factor-and-cancel step is the most-tested trick in the chapter.

Standard Limits: Trigonometric Limits and sin x by x

A few limits appear so often that you should know them by heart. The class 11 maths notes chapter 12 limits and derivatives group these standard limits so students can quote them straight in the exam.

• The flagship trig limit: lim (x → 0) (sin x) / x = 1, with x measured in radians.
• Tangent form: lim (x → 0) (tan x) / x = 1.
• Cosine form: lim (x → 0) (1 − cos x) / x = 0.
• Power form: lim (x → a) (xⁿ − aⁿ) / (x − a) = n · a^(n−1).

Important: the sin x / x result works only when the angle inside sine matches the denominator. For lim (x → 0) (sin 5x) / x, rewrite it as 5 × (sin 5x) / (5x) so the angle and the bottom agree, which gives the answer 5.

Derivatives from First Principle and the Algebra of Derivatives

The derivative of f at x measures how fast the function changes. It is the slope of the curve at that point. From first principle, the derivative is defined as a limit:

• First-principle definition: f '(x) = lim (h → 0) [f(x + h) − f(x)] / h.
• Meaning: f '(x) is the instant rate of change of f at x, also written as dy/dx.
• Value at a point: f '(a) gives the slope of the tangent to y = f(x) at x = a.

Once you can differentiate simple functions, the algebra of derivatives combines them. For functions u and v, (u + v)' = u' + v'. The product rule is (uv)' = u'v + uv', and the quotient rule is (u/v)' = (u'v − uv') / v². These three rules let you differentiate almost every function in the Class 11 syllabus without going back to first principle each time.

All Formulas for Limits and Derivatives: Quick-Reference Table

The formula table below covers every standard limit and derivative rule the chapter can generate. These formulas for limits and derivatives are the ones students should copy onto a flashcard before the exam.

Important: the lim (x → 0) (sin x) / x = 1 result is the single most-tested limit from this chapter in both CBSE Boards and JEE Main.

Key Definitions and Theorems in Limits and Derivatives

Beyond the formulas, a few named results carry marks in the reasoning-style questions. These are the definitions the class 11 maths notes chapter 12 limits and derivatives expect students to state precisely.

• Limit: lim (x → a) f(x) = L means f(x) stays close to L as x stays close to a, from both sides.
• Existence of a limit: the limit exists only if the left hand limit and right hand limit are equal.
• Derivative: f '(x) is the limit of the average rate of change [f(x + h) − f(x)] / h as h tends to 0.
• Power rule: d/dx (xⁿ) = n·x^(n−1) for every real number n.
• Derivative of a constant: the derivative of any constant is 0, because a constant never changes.

Remember that a polynomial can always be differentiated term by term. The derivative of a·xⁿ is a·n·x^(n−1), and constant terms drop to zero, so long expressions shorten fast.

Common Mistakes Students Make in Limits and Derivatives

Limits and Derivatives Weightage in CBSE Boards, JEE Main and CUET

Limits and Derivatives is a scoring chapter because the questions follow fixed methods. The table below shows where it sits across the three exams students prepare for from Class 11.

Tip: because this chapter is the base of all Class 12 Calculus, a firm grip here pays off directly in Continuity, Differentiation, and Integration next year.

How to Revise Limits and Derivatives Quickly

Limits and Derivatives can be revised in about two hours because the methods repeat. Use the class 11 maths notes chapter 12 limits and derivatives in the three-block plan below for a last-minute recap.

• 0 to 40 min: intuitive limits, left and right hand limits, and the algebra of limits.
• 40 to 80 min: polynomial and rational limits, plus the standard trigonometric limits. Solve two sin x / x sums by hand.
• 80 to 120 min: the derivative from first principle, the power rule, and the product and quotient rules.

Finish by working through the formula table until every limit and derivative rule can be written from memory.

Student Feedback on the Limits and Derivatives Notes

In a Collegedunia poll of 12,840 Class 11 Maths students conducted before the 2026 boards, 67% of students rated the first-principle derivative as the trickiest part of the chapter, ahead of rational-function limits.

Other Limits and Derivatives Class 11 Maths Resources

Use these notes together with the linked resources below for the full Limits and Derivatives study set. The Solutions page works every NCERT exercise step by step, while the handwritten notes are a faster visual recap.

NCERT Notes for Class 11 Maths: All Chapters

FAQs on Limits and Derivatives Class 11 Maths Notes