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.
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…
- + 5 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 |
Showing 3 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)
...
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) (...
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(...
Showing 3 of 8 questions. Visit the full page for complete solutions.