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Limits and Derivatives

NCERT Class 11 Mathematics Chapter 12

Limits and Derivatives for Class 11 Maths and JEE: limits, algebra of limits, trigonometric limits, the derivative from first principles and its rules.

4 topics Updated 8 Nov 2026

For: Class 11 and JEE

In the syllabus: JEE Class 11 syllabus

With your textbook

NCERT Mathematics (Class 11), Chapter 12 “Limits and Derivatives”, reprint 2026-27.

Open the official NCERT chapter (PDF, 3.4 MB, ncert.nic.in)

Checked on 19 Oct 2026.

Learn in this order

  1. Intuitive idea of derivatives

    The intuitive idea of derivatives via average velocity, secants and the tangent, with solved examples.

    NCERT Sections 12.1–12.2 13 min read 6 solved examples 12 practice questions

  2. Limits

    Limit of a function, left- and right-hand limits, algebra of limits, 0/0 forms and a formula sheet, with solved examples.

    NCERT Section 12.3 18 min read 8 solved examples 16 practice questions

  3. Limits of trigonometric functions

    The sandwich theorem, the proof that sin x/x tends to 1, the standard limits and solved examples.

    NCERT Section 12.4 14 min read 6 solved examples 12 practice questions

  4. Derivatives

    First principle, slope of the tangent, product and quotient rules, formulas for x, sin x and cos x, and solved examples.

    NCERT Section 12.5 16 min read 8 solved examples 22 practice questions

Key formulas in this chapter

Intuitive idea of derivatives

  • average velocity = (s(t) − s(t))/(t − t)
  • s = 4.9t: (s(2 + h) − s(2))/h = 19.6 + 4.9h → 19.6
  • derivative at a: (f(a + h) − f(a))/h as h → 0
  • slope of tangent = limit of slopes of chords

Limits

  • limit exists ⟺ left-hand limit = right-hand limit
  • lim (f ± g) = lim f ± lim g; lim fg = lim f · lim g
  • lim (f/g) = lim f / lim g, if lim g ≠ 0
  • polynomial p: lim (x→a) p(x) = p(a)
  • lim (x→a) (x − a)/(x − a) = na (n a positive integer, or n rational and a > 0)

Limits of trigonometric functions

  • cos x < (sin x)/x < 1 for 0 < |x| < π/2
  • lim (x→0) (sin x)/x = 1
  • lim (x→0) (1 − cos x)/x = 0
  • lim (x→0) (tan x)/x = 1
  • lim (x→0) (sin ax)/(bx) = a/b, b ≠ 0

Derivatives

  • f′(x) = lim (h→0) (f(x + h) − f(x))/h
  • (uv)′ = u′v + uv′; (u/v)′ = (u′v − uv′)/v, v ≠ 0
  • (x)′ = nx (real n: x > 0; integer powers on their domains)
  • (sin x)′ = cos x; (cos x)′ = −sin x; (tan x)′ = sec x
  • (cot x)′ = −cosec x; (sec x)′ = sec x tan x; (cosec x)′ = −cosec x cot x

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