Limits and Derivatives — Class 11 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 11 Mathematics chapter "Limits and Derivatives" — 8 important questions with detailed answers for CBSE board exam preparation.
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TL;DR: Free step-by-step NCERT solutions for Class 11 Mathematics chapter "Limits and Derivatives" — 8 important questions with detailed answers for CBSE boa…
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Key Questions Covered:
- Find the limit: lim(x→2) (x² - 4)/(x - 2)
- Find the limit: lim(x→0) (sin x)/x
- Find the derivative of f(x) = 3x² + 2x - 5 using the first principle of deriv…
- Find the derivative of f(x) = √(2x + 3)
- Find the derivative of f(x) = x² sin x
- Find the derivative of f(x) = (x² + 1)/(x - 1)
- + 2 more questions in the full chapter
Solutions Summary:
| Question | Status |
|---|---|
| Find the limit: lim(x→2) (x² - 4)/(x - 2) | ✓ Solved |
| Find the limit: lim(x→0) (sin x)/x | ✓ Solved |
| Find the derivative of f(x) = 3x² + 2x - 5 using the firs… | ✓ Solved |
| Find the derivative of f(x) = √(2x + 3) | ✓ Solved |
| Find the derivative of f(x) = x² sin x | ✓ Solved |
| Find the derivative of f(x) = (x² + 1)/(x - 1) | ✓ Solved |
Showing 6 of 8 questions
Q1: Find the limit: lim(x→2) (x² - 4)/(x - 2)
Find: lim(x→2) (x² - 4)/(x - 2)
Step 1: Check for direct substitution:
When x = 2: (2² - 4)/(2 - 2) = 0/0 (indeterminate form)
Step 2: Factorize the numerator:
x² - 4 = (x + 2)(x - 2)
Step 3: Cancel common factors:
lim(x→2) (x² - 4)/(x - 2) = lim(x→2) [(x + 2)(x - 2)]/(x - 2)
= lim(x→2) (x + 2)
Step 4: Apply direct substitution:
= 2 + 2 = 4
Final Answer: The limit is 4
Q2: Find the limit: lim(x→0) (sin x)/x
Find: lim(x→0) (sin x)/x
Step 1: This is a standard limit form.
Step 2: Direct substitution gives 0/0 (indeterminate).
Step 3: Use L'Hôpital's Rule:
lim(x→0) (sin x)/x = lim(x→0) (d/dx sin x)/(d/dx x)
= lim(x→0) (cos x)/1
= cos 0 = 1
Alternatively, this is a well-known standard limit:
lim(x→0) (sin x)/x = 1
Final Answer: The limit is 1
Q3: Find the derivative of f(x) = 3x² + 2x - 5 using the first principle of derivatives.
Find the derivative of f(x) = 3x² + 2x - 5 using first principle.
Step 1: Use the definition:
f'(x) = lim(h→0) [f(x + h) - f(x)]/h
Step 2: Calculate f(x + h):
f(x + h) = 3(x + h)² + 2(x + h) - 5
= 3(x² + 2xh + h²) + 2x + 2h - 5
= 3x² + 6xh + 3h² + 2x + 2h - 5
Step 3: Calculate f(x + h) - f(x):
f(x + h) - f(x) = [3x² + 6xh + 3h² + 2x + 2h - 5] - [3x² + 2x - 5]
= 6xh + 3h² + 2h
= h(6x + 3h + 2)
Step 4: Apply the limit:
f'(x) = lim(h→0) [h(6x + 3h + 2)]/h
= lim(h→0) (6x + 3h + 2)
= 6x + 0 + 2
=...
Q4: Find the derivative of f(x) = √(2x + 3)
Find the derivative of f(x) = √(2x + 3)
Step 1: Rewrite the function:
f(x) = (2x + 3)^(1/2)
Step 2: Use the chain rule:
f'(x) = d/dx[(2x + 3)^(1/2)]
= (1/2)(2x + 3)^(-1/2) × d/dx(2x + 3)
Step 3: Find the derivative of the inner function:
d/dx(2x + 3) = 2
Step 4: Substitute:
f'(x) = (1/2)(2x + 3)^(-1/2) × 2
= (2x + 3)^(-1/2)
= 1/√(2x + 3)
Final Answer: f'(x) = 1/√(2x + 3)
Q5: Find the derivative of f(x) = x² sin x
Find the derivative of f(x) = x² sin x
Step 1: This is a product of two functions.
Let u = x² and v = sin x
Step 2: Use the product rule:
f'(x) = u'v + uv'
Step 3: Find u' and v':
u' = 2x
v' = cos x
Step 4: Apply the product rule:
f'(x) = (2x)(sin x) + (x²)(cos x)
= 2x sin x + x² cos x
= x(2 sin x + x cos x)
Final Answer: f'(x) = 2x sin x + x² cos x
Q6: Find the derivative of f(x) = (x² + 1)/(x - 1)
Find the derivative of f(x) = (x² + 1)/(x - 1)
Step 1: This is a quotient of two functions.
Let u = x² + 1 and v = x - 1
Step 2: Use the quotient rule:
f'(x) = [u'v - uv']/v²
Step 3: Find u' and v':
u' = 2x
v' = 1
Step 4: Apply the quotient rule:
f'(x) = [(2x)(x - 1) - (x² + 1)(1)]/(x - 1)²
= [2x² - 2x - x² - 1]/(x - 1)²
= [x² - 2x - 1]/(x - 1)²
Final Answer: f'(x) = (x² - 2x - 1)/(x - 1)²
Showing 6 of 8 questions. Visit the full page for complete solutions.
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