The limit of (x² − 4)/(x − 2) as x tends to 2 is 4 and the limit of (sin x)/x as x tends to 0 is 1, so this Class 11 sheet opens with the two standard limits. The derivative answers record 2x, 0, 6 and cos x.
Key Takeaways
lim (x → 2) (x² − 4)/(x − 2) = 4 (Q1).
lim (x → 0) (sin x)/x = 1 (Q2).
Q6 records cos x as a derivative result.
Section A: Multiple Choice Questions (1 Mark Each)
Choose the correct option for each question.
1.lim (x → 2) (x² - 4)/(x - 2) = ?
(a) 2(b) 4(c) 0(d) Not defined
2.lim (x → 0) (sin x)/x = ?
(a) 0(b) 1(c) ∞(d) -1
3.The derivative of x² with respect to x is:
(a) 2x(b) x(c) 2(d) x²/2
4.The derivative of the constant function f(x) = 7 is:
(a) 7(b) 1(c) 0(d) -7
5.lim (x → 3) (x² - 9)/(x - 3) = ?
(a) 3(b) 6(c) 9(d) 0
6.The derivative of sin x with respect to x is:
(a) -cos x(b) cos x(c) -sin x(d) tan x
Section B: Short Answer Type Questions (2 Marks Each)
Show all steps clearly.
7.State the value of lim (x → 0) (cos x - 1)/x.
8.Find lim (x → 2) (3x + 1).
9.Find lim (x → 0) (tan x)/x.
10.Find the derivative of x³ with respect to x.
11.Find the derivative of 5x - 3 with respect to x.
12.Find lim (x → 1) (x² - 1)/(x - 1).
13.Find the derivative of x⁵ with respect to x.
14.If f(x) = 3x² + 2x, find f′(x).
Section C: Numericals & Word Problems (3 Marks Each)
Apply the concepts to solve the problems. Show all working.
15.The distance covered by a particle is s(t) = t² + 2t metres after t seconds. Find the velocity of the particle at t = 3 seconds.
16.A firm has cost C(x) = 100 + 20x and revenue R(x) = 30x. Find the marginal revenue, which is the derivative of R(x).
17.The position of a body is given by s(t) = t³ - 3t. Find its velocity at t = 2 seconds.
18.The area of a square with side x is A(x) = x². Find the rate of change of the area when x = 5.
19.A stone falls freely so that the distance fallen is s = 5t² metres after t seconds. Find the velocity of the stone at t = 2 seconds.
20.If y = 1/x, find the value of dy/dx at x = 2.
Answer Key
1. b) 4
2. b) 1
3. a) 2x
4. c) 0
5. b) 6
6. b) cos x
7. Refer to solution guide.
8. Refer to solution guide.
9. Refer to solution guide.
10. Refer to solution guide.
11. Refer to solution guide.
12. Refer to solution guide.
13. Refer to solution guide.
14. Refer to solution guide.
15. Refer to solution guide.
16. Refer to solution guide.
17. Refer to solution guide.
18. Refer to solution guide.
19. Refer to solution guide.
20. Refer to solution guide.
End of Worksheet
People Also Ask
What is the limit of (x² − 4)/(x − 2) as x tends to 2?
4, option (b) in this worksheet (Q1).
What is the limit of (sin x)/x as x tends to 0?
1, option (b) in this worksheet (Q2).
What is the derivative of sin x?
cos x, option (b) in this worksheet (Q6).
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