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CBSE · Class 11 · Mathematics

Limits and Derivatives

Standard difficulty · 20 questions · 40 marks

Name: Date: Score: ____ / 40

Show your working for Sections B and C. Answers are on the last page — try every question first!

Section A — Choose the correct answer8 × 1 = 8 marks
1.
lim (x → 3) (3x2 + x + 3) equals:
  1. −33
  2. 34
  3. 27
  4. 33
2.
lim (x → 4) (x6 − 4096) ÷ (x − 4) equals:
  1. 4,096
  2. 24,576
  3. 6,144
  4. 5,120
3.
lim (x → 0) (sin 2x ÷ sin 2x) equals:
  1. 2
  2. 3
  3. 0
  4. 1
4.
The derivative of 4x4 − 4x3 − x2 − 3x − 2 is:
  1. 16x3 − 12x2 − 2x − 5
  2. 16x3 − 12x2 − 2x − 3
  3. 4x3 − 4x2 − x − 3
  4. 20x3 − 12x2 − 2x − 3
5.
lim (x → 0) (sin x)/x equals:
  1. it does not exist
  2. 0
  3. 1
  4. ∞
6.
If f(x) = −4x3 + 3x2 + 4x + 4, then f′(1) is:
  1. 2
  2. 7
  3. −2
  4. −1
7.
The derivative of −3 sin x − cos x − x5 is:
  1. −3 cos x + sin x − 5x4
  2. 3 cos x + sin x − 5x4
  3. −3 cos x − sin x − 5x4
  4. −3 cos x + sin x − x4
8.
Assertion (A): The derivative of x8 is x7.
Reason (R): For any positive integer n, the derivative of xⁿ is nxⁿ−1.
  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, but R is not the correct explanation of A.
  3. A is true, but R is false.
  4. A is false, but R is true.
Section B — Short answer8 × 2 = 16 marks
9.
Evaluate lim (x → 3) (x2 − 5x + 6) ÷ (x − 3).
10.
Evaluate lim (x → 3) (x3 − 27) ÷ (x − 3).
11.
Evaluate lim (x → 0) (sin 4x + 3x) ÷ x.
12.
Find the derivative of 5x5 − 6x4 + 4x3 + 2x2 − x.
13.
Find the derivative of (−2x − 3)(x2 + 1) using the product rule.
14.
Find the derivative of (−6x + 2) ÷ (3x + 2).
15.
Find the derivative of f(x) = 4x2 + x + 8 from first principles, and hence find f′(3).
16.
Differentiate x6 cos x.
Section C — Word problems4 × 4 = 16 marks
17.
Evaluate: (a) lim (x → 4) (x6 − 4,096) ÷ (x − 4); (b) lim (x → 0) (sin 3x) ÷ (sin 4x); (c) lim (x → −1) (x2 − 1) ÷ (x + 1).
18.
f(x) = −x + 1 for x < −3 and f(x) = 4x + 16 for x ≥ −3. Find the left-hand and right-hand limits at x = −3. Does lim (x → −3) f(x) exist?
19.
For f(x) = −2x3 + 1, find f′(x), f′(3) and f′(0). What is the slope of the tangent to the graph at x = 3?
20.
Find the derivative of f(x) = 1 ÷ (x + 1) from first principles.
AmbifulWorksheets
Practice that never repeats
CBSE · Class 11 · Mathematics

Answer key

Limits and Derivatives · AW-11M-LMD-2026-V1 · For parents — fold back or tear off before handing the sheet to your child.

QAnswerWorking
1(d) 33substitute directly
2(c) 6,144naⁿ−1 = 6 × 45
3(d) 1use lim (sin θ)/θ = 1
4(b) 16x3 − 12x2 − 2x − 3d/dx xⁿ = nxⁿ−1
5(c) 1x in radians
6(c) −2f′(x) = −12x2 + 6x + 4
7(a) −3 cos x + sin x − 5x4(sin x)′ = cos x, (cos x)′ = −sin x
8(d) A is false, but R is true.A is false: d/dx xⁿ = nxⁿ−1 with n = 8. R is true.
91factorise: (x − 3)(x − 2); cancel and put x = 3
1027lim (xⁿ − aⁿ)/(x − a) = naⁿ−1 = 3 × 32
117sin 4x ÷ x → 4; 3x ÷ x = 3
1225x4 − 24x3 + 12x2 + 4x − 1differentiate term by term
13−6x2 − 6x − 2u′v + uv′ = −2(x2 + 1) + (−2x − 3)(2x)
14−18 ÷ (3x + 2)2(u′v − uv′)/v2 = [−6(3x + 2) − 3(−6x + 2)] ÷ (3x + 2)2
15f′(x) = 8x + 1; f′(3) = 25[f(x + h) − f(x)] ÷ h = 8x + 1 + 4h → 8x + 1 as h → 0
166x5 cos x − x6 sin xproduct rule
17(a) 6,144; (b) 3/4; (c) −2(c): x2 − r2 = (x − r)(x + r)
18LHL = 4, RHL = 4; yes, the limit is 4put x = −3 in each expression
19f′(x) = −6x2; f′(3) = −54; f′(0) = 0; the slope is −54—
20f′(x) = −1 ÷ (x + 1)2[1/(x + h + 1) − 1/(x + 1)] ÷ h = −1 ÷ [(x + h + 1)(x + 1)] → −1/(x + 1)2