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Chapter 13: Limits & Derivatives

NCERT Solutions for CBSE Class 10 Mathematics — 13 solved questions with detailed explanations.

13

Questions

0

Topics

Solved Questions

Q1. lim(x→0) sin(x)/x equals:

0
1
∞
Does not exist

Difficulty: Easy · Topic: Standard Limits

Solution

This is a fundamental limit: lim(x→0) sin(x)/x = 1.

Q2. The derivative of xⁿ is:

xⁿ⁻¹
nxⁿ⁻¹
nxⁿ
xⁿ/n

Difficulty: Easy · Topic: Differentiation

Solution

Power rule: d/dx(xⁿ) = nxⁿ⁻¹.

Q3. The derivative of sin(x) is:

-sin(x)
cos(x)
-cos(x)
sin(x)

Difficulty: Easy · Topic: Trig Derivatives

Solution

d/dx[sin(x)] = cos(x).

Q4. lim(x→1) (x²-1)/(x-1) equals:

0
1
2
Does not exist

Difficulty: Easy-Medium · Topic: Limits

Solution

(x²-1)/(x-1) = (x+1)(x-1)/(x-1) = x+1. lim(x→1) = 2.

Q5. d/dx(e^x) =

1
e^x
e^x/x
xe^(x-1)

Difficulty: Easy-Medium · Topic: Limits & Derivatives

Solution

e^x.

Q6. d/dx(ln x) =

ln x
1/x
e^x
x

Difficulty: Easy-Medium · Topic: Limits & Derivatives

Solution

1/x.

Q7. d/dx(cos x) =

-sin x
sin x
cos x
-cos x

Difficulty: Easy-Medium · Topic: Limits & Derivatives

Solution

-sin x.

Q8. d/dx(tan x) =

-cosec²x
sec²x
-sec²x
cosec²x

Difficulty: Easy-Medium · Topic: Limits & Derivatives

Solution

sec²x.

Q9. d/dx(xⁿ) =

xⁿ
x^(n+1)/(n+1)
nx^(n-1)
nxⁿ

Difficulty: Easy-Medium · Topic: Limits & Derivatives

Solution

nx^(n-1).

Q10. lim(x→0) (e^x - 1)/x =

∞
e
1
0

Difficulty: Easy-Medium · Topic: Limits & Derivatives

Solution

1.

Q11. lim(x→∞) (1+1/x)^x =

1
0
e
∞

Difficulty: Easy-Medium · Topic: Limits & Derivatives

Solution

e.

Q12. Product rule: d/dx(uv) =

(uv)' = u'v'
u'v'
uv' + u'v'
u'v + uv'

Difficulty: Easy-Medium · Topic: Limits & Derivatives

Solution

u'v + uv'.

Q13. Quotient rule: d/dx(u/v) =

(u'v + uv')/v²
u'v' - uv
(u'v - uv')/v²
u'/v'

Difficulty: Easy-Medium · Topic: Limits & Derivatives

Solution

(u'v - uv')/v².

Other Chapters in Mathematics

Ch 1: Sets Ch 2: Relations & Functions Ch 3: Trigonometric Functions Ch 4: Principle of Mathematical Induction Ch 5: Complex Numbers Ch 6: Linear Inequalities Ch 7: Permutations & Combinations Ch 8: Binomial Theorem Ch 9: Sequences & Series Ch 10: Straight Lines Ch 11: Conic Sections Ch 12: Intro to 3D Geometry Ch 14: Mathematical Reasoning Ch 15: Statistics Ch 16: Probability

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