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Solutions

Overview

Step-by-step NCERT solutions for Limits and Derivatives (Chapter 12, NCERT Class 11 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Limits and Derivatives textbook chapter.

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All 73 questions in Limits and Derivatives are solved in the PDF. Here's what's inside, exercise by exercise:

Exercise 12.1

  1. Evaluate: lim(x→3) (x + 3)
  2. Evaluate: lim(x→π) (x - 22/7)
  3. Evaluate: lim(r→1) πr²
  4. Evaluate: lim(x→4) (4x + 3)/(x - 2)
  5. Evaluate: lim(x→-1) (x¹⁰ + x⁵ + 1)/(x - 1)
  6. Evaluate: lim(x→0) [(x + 1)⁵ - 1] / x
  7. Evaluate: lim(x→2) (3x² - x - 10)/(x² - 4)
  8. Evaluate: lim(x→3) (x⁴ - 81)/(2x² - 5x - 3)
  9. Evaluate: lim(x→0) (ax + b)/(cx + 1)
  10. Evaluate: lim(z→1) (z^(1/3) - 1)/(z^(1/6) - 1)
  11. Evaluate: lim(x→1) (ax² + bx + c)/(cx² + bx + a), where a + b + c ≠ 0
  12. Evaluate: lim(x→-2) [(1/x + 1/2)/(x + 2)]
  13. Evaluate: lim(x→0) sin(ax)/(bx)
  14. Evaluate: lim(x→0) sin(ax)/sin(bx), where a, b ≠ 0
  15. Evaluate: lim(x→π) sin(π - x)/[π(π - x)]
  16. Evaluate: lim(x→0) cos x/(π - x)
  17. Evaluate: lim(x→0) (cos 2x - 1)/(cos x - 1)
  18. Evaluate: lim(x→0) (ax + x cos x)/(b sin x)
  19. Evaluate: lim(x→0) x sec x
  20. Evaluate: lim(x→0) (sin ax + bx)/(ax + sin bx), where a, b, a + b ≠ 0
  21. Evaluate: lim(x→0) (cosec x - cot x)
  22. Evaluate: lim(x→π/2) tan 2x / (x - π/2)
  23. Find lim(x→0) f(x) and lim(x→1) f(x), where f(x) = 2x + 3 for x ≤ 0, and f(x) = 3(x + 1) for x > 0.
  24. Find lim(x→1) f(x), where f(x) = x² - 1 for x ≤ 1, and f(x) = -x² - 1 for x > 1.
  25. Evaluate lim(x→0) f(x), where f(x) = |x|/x for x ≠ 0 and f(0) = 0.
  26. Find lim(x→0) f(x), where f(x) = x/|x| for x ≠ 0 and f(0) = 0.
  27. Find lim(x→5) f(x), where f(x) = |x| - 5.
  28. Suppose f(x) = a + bx for x < 1, f(1) = 4, and f(x) = b - ax for x > 1. If lim(x→1) f(x) = f(1), what are the possible values of a and b?
  29. Let a₁, a₂, ..., aₙ be fixed real numbers and define f(x) = (x - a₁)(x - a₂)⋯(x - aₙ). What is lim(x→a₁) f(x)? For some a ≠ a₁, a₂, ..., aₙ, compute lim(x→a) f(x).
  30. If f(x) = x + 1 for x < 0, f(0) = 0, and f(x) = x - 1 for x > 0, for what value(s) of a does lim(x→a) f(x) exist?
  31. If the function f(x) satisfies lim(x→1) [f(x) - 2]/(x² - 1) = π, evaluate lim(x→1) f(x).
  32. If f(x) = mx² + n for x < 0, f(x) = nx + m for 0 ≤ x ≤ 1, and f(x) = nx³ + m for x > 1, for what integers m and n do both lim(x→0) f(x) and lim(x→1) f(x) exist?

Exercise 12.2

  1. Find the derivative of x² - 2 at x = 10.
  2. Find the derivative of x at x = 1.
  3. Find the derivative of 99x at x = 100.
  4. Find the derivative of the following functions from first principle:
    • (i) x³ - 27
    • (ii) (x - 1)(x - 2)
    • (iii) 1/x²
    • (iv) (x + 1)/(x - 1)
  5. For the function f(x) = x¹⁰⁰/100 + x⁹⁹/99 + … + x²/2 + x + 1, prove that f'(1) = 100 f'(0).
  6. Find the derivative of x^n + ax^(n-1) + a²x^(n-2) + … + a^(n-1)x + a^n for some fixed real number a.
  7. For some constants a and b, find the derivative of:
    • (i) (x - a)(x - b)
    • (ii) (ax² + b)²
    • (iii) (x - a)/(x - b)
  8. Find the derivative of (x^n - a^n)/(x - a) for some constant a.
  9. Find the derivative of:
    • (i) 2x - 3/4
    • (ii) (5x³ + 3x - 1)(x - 1)
    • (iii) x^(-3)(5 + 3x)
    • (iv) x⁵(3 - 6x^(-9))
    • (v) x^(-4)(3 - 4x^(-5))
    • (vi) 2/(x+1) - x²/(3x-1)
  10. Find the derivative of cos x from first principle.
  11. Find the derivative of the following functions:
    • (i) sin x cos x
    • (ii) sec x
    • (iii) 5 sec x + 4 cos x
    • (iv) cosec x
    • (v) 3 cot x + 5 cosec x
    • (vi) 5 sin x - 6 cos x + 7
    • (vii) 2 tan x - 7 sec x

Miscellaneous Exercise

  1. Find the derivative of the following functions from first principle:
    • (i) -x
    • (ii) (-x)^(-1)
    • (iii) sin(x + 1)
    • (iv) cos(x - π/8)
  2. Find the derivative of (x + a).
  3. Find the derivative of (px + q)(r/x + s).
  4. Find the derivative of (ax + b)(cx + d)².
  5. Find the derivative of (ax + b)/(cx + d).
  6. Find the derivative of (1 + 1/x)/(1 - 1/x).
  7. Find the derivative of 1/(ax² + bx + c).
  8. Find the derivative of (ax + b)/(px² + qx + r).
  9. Find the derivative of (px² + qx + r)/(ax + b).
  10. Find the derivative of a/x⁴ - b/x² + cos x.
  11. Find the derivative of 4√x - 2.
  12. Find the derivative of (ax + b)^n.
  13. Find the derivative of (ax + b)^n (cx + d)^m.
  14. Find the derivative of sin(x + a).
  15. Find the derivative of cosec x cot x.
  16. Find the derivative of cos x/(1 + sin x).
  17. Find the derivative of (sin x + cos x)/(sin x - cos x).
  18. Find the derivative of (sec x - 1)/(sec x + 1).
  19. Find the derivative of sin^n x.
  20. Find the derivative of (a + b sin x)/(c + d cos x).
  21. Find the derivative of sin(x + a)/cos x.
  22. Find the derivative of x⁴(5 sin x - 3 cos x).
  23. Find the derivative of (x² + 1) cos x.
  24. Find the derivative of (ax + sin x)(p + q cos x).
  25. Find the derivative of (x + cos x)(x - tan x).
  26. Find the derivative of (4x + 5 sin x)/(3x + 7 cos x).
  27. Find the derivative of x² cos(π/4)/sin x.
  28. Find the derivative of x/(1 + tan x).
  29. Find the derivative of (x + sec x)(x - tan x).
  30. Find the derivative of x/sin^n x.
Sample

Worked examples

3 questions from Limits and Derivatives, solved step by step.

Exercise 12.1

Evaluate: lim(x→3) (x + 3)

The function f(x) = x + 3 is a polynomial.

For any polynomial, the limit as x → a equals the value of the polynomial at x = a.

lim(x→3) (x + 3) = 3 + 3 = 6

Answer: 6

Exercise 12.2

Find the derivative of x² - 2 at x = 10.

Let f(x) = x² - 2. Using the definition of derivative:

f'(x) = lim(h→0) [f(x+h) - f(x)]/h = lim(h→0) [(x+h)² - 2 - (x²-2)]/h

= lim(h→0) [x²+2xh+h²-x²]/h = lim(h→0) (2x+h) = 2x.

At x = 10: f'(10) = 2(10) = 20.

Answer: 20

Miscellaneous Exercise

Find the derivative of (px + q)(r/x + s).

Expand: (px + q)(r/x + s) = pr + psx + qr/x + qs.

Differentiate term by term:

d/dx(pr) = 0, d/dx(psx) = ps, d/dx(qr/x) = -qr/x², d/dx(qs) = 0.

f'(x) = ps - qr/x².

Answer: ps - qr/x²

All 73 questions in Limits and Derivatives are solved step by step in the Solutions PDF above.

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