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Limits and Derivatives - Important Questions

⏱️ Est. Revision Time: 15 Mins
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1
Exercise 13.1
Evaluate: lim (x → 2) (x² - 4)/(x - 2).
▶ Show Detailed Solution
Factor numerator:
(x² - 4)/(x - 2) = (x - 2)(x + 2)/(x - 2) = x + 2

lim (x → 2) (x + 2) = 4
2
Exercise 13.1
Evaluate: lim (x → 0) sin(3x)/x.
▶ Show Detailed Solution
Use standard limit: lim (x→0) sin(kx)/(kx) = 1.
lim (x→0) sin(3x)/x = lim (x→0) [3 · sin(3x)/(3x)]
= 3 × 1 = 3
3
Exercise 13.1
Evaluate: lim (x → 0) (1 - cos x)/x².
▶ Show Detailed Solution
Use identity: 1 - cos x = 2sin²(x/2).
= lim (x→0) 2sin²(x/2)/x²
= lim (x→0) 2 · [sin(x/2)/(x/2)]² · (1/4)
= 2 × 1 × 1/4 = 1/2
4
Exercise 13.1
Find the derivative of f(x) = x³ from first principles.
▶ Show Detailed Solution
f'(x) = lim (h→0) [f(x+h) - f(x)]/h
= lim (h→0) [(x+h)³ - x³]/h
= lim (h→0) [3x²h + 3xh² + h³]/h
= lim (h→0) [3x² + 3xh + h²]
= 3x²
5
Exercise 13.1
Find the derivative of f(x) = sin x from first principles.
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f'(x) = lim (h→0) [sin(x+h) - sin x]/h
= lim (h→0) [2cos(x + h/2) sin(h/2)]/h
= lim (h→0) cos(x + h/2) · [sin(h/2)/(h/2)]
= cos x × 1 = cos x
6
Exercise 13.1
Differentiate f(x) = x³ + 2x² - 5x + 7 using the power rule.
▶ Show Detailed Solution
d/dx[xⁿ] = nxⁿ⁻¹
f'(x) = 3x² + 4x - 5