Limits
A limit finds the value a function approaches as the input gets closer to a point.
Key Formulas: - lim(x→a) c = c - lim(x→a) x = a - lim(x→a) (f(x) ± g(x)) = lim f(x) ± lim g(x) - lim(x→a) (f(x) * g(x)) = lim
f(x) * lim g(x) - lim(x→a) f(x)/g(x) = lim f(x) / lim g(x), if denominator ≠ 0
Standard Limits: lim(x→0) (sin x)/x = 1, lim(x→0) (1 - cos x)/x² = 1/2, lim(x→0) (e^x - 1)/x = 1
Example: Find lim(x→0) (sin 3x)/x. Solution: lim(sin 3x / x) = (sin 3x / 3x) * 3 = 3.
Tips: - Factorize or rationalize if direct substitution gives 0/0. - Use L'Hospital's Rule for 0/0 or ∞/∞ forms.
Continuity & Differentiability
Continuity: A function f(x) is continuous at x=a if: 1. f(a) is defined 2. lim(x→a) f(x) exists 3. lim(x→a) f(x) = f(a)
Differentiability: A function is differentiable at x=a if left-hand derivative = right-hand derivative.
Example: Check continuity of f(x) = (x² - 1)/(x - 1) at x=1. Solution: For x≠1, f(x)=x+1. Limit at x=1 = 2, but f(1) not
defined → Not continuous.
Derivatives
Definition: f'(x) = lim(h→0) [f(x+h) - f(x)] / h
Rules: - Power Rule: d/dx(x^n) = n x^(n-1) - Sum Rule: (u+v)' = u' + v' - Product Rule: (uv)' = u'v + uv' - Quotient Rule:
(u/v)' = (u'v - uv')/v² - Chain Rule: dy/dx = (dy/du)(du/dx)
Common Derivatives: d/dx(sin x) = cos x, d/dx(cos x) = -sin x, d/dx(e^x) = e^x, d/dx(ln x) = 1/x
Example: Differentiate y = x² e^x. Solution: dy/dx = e^x(2x + x²).
Applications of Derivatives
Increasing/Decreasing: f'(x) > 0 → increasing, f'(x) < 0 → decreasing.
Maxima/Minima: f'(x) = 0 → critical point. If f''(x) > 0 → local minimum, f''(x) < 0 → local maximum.
Example: f(x) = x² - 4x + 3. f'(x) = 2x - 4 = 0 → x=2. f''(x)=2>0 → Minimum at x=2, f(2)=-1.
Integrals
Definition: Integration is reverse of differentiation.
Standard Results: ∫ x^n dx = x^(n+1)/(n+1) + C (n≠-1) ∫ (1/x) dx = ln|x| + C ∫ e^x dx = e^x + C ∫ sin x dx = -cos x + C
Methods: 1. Substitution 2. By Parts: ∫ u v dx = u∫ v dx - ∫(du/dx ∫ v dx) dx 3. Partial Fractions
Example: ∫ x sin x dx = -x cos x + sin x + C.
Definite Integrals
A limit finds the value a function approaches as the input gets closer to a point.
Key Formulas: - lim(x→a) c = c - lim(x→a) x = a - lim(x→a) (f(x) ± g(x)) = lim f(x) ± lim g(x) - lim(x→a) (f(x) * g(x)) = lim
f(x) * lim g(x) - lim(x→a) f(x)/g(x) = lim f(x) / lim g(x), if denominator ≠ 0
Standard Limits: lim(x→0) (sin x)/x = 1, lim(x→0) (1 - cos x)/x² = 1/2, lim(x→0) (e^x - 1)/x = 1
Example: Find lim(x→0) (sin 3x)/x. Solution: lim(sin 3x / x) = (sin 3x / 3x) * 3 = 3.
Tips: - Factorize or rationalize if direct substitution gives 0/0. - Use L'Hospital's Rule for 0/0 or ∞/∞ forms.
Continuity & Differentiability
Continuity: A function f(x) is continuous at x=a if: 1. f(a) is defined 2. lim(x→a) f(x) exists 3. lim(x→a) f(x) = f(a)
Differentiability: A function is differentiable at x=a if left-hand derivative = right-hand derivative.
Example: Check continuity of f(x) = (x² - 1)/(x - 1) at x=1. Solution: For x≠1, f(x)=x+1. Limit at x=1 = 2, but f(1) not
defined → Not continuous.
Derivatives
Definition: f'(x) = lim(h→0) [f(x+h) - f(x)] / h
Rules: - Power Rule: d/dx(x^n) = n x^(n-1) - Sum Rule: (u+v)' = u' + v' - Product Rule: (uv)' = u'v + uv' - Quotient Rule:
(u/v)' = (u'v - uv')/v² - Chain Rule: dy/dx = (dy/du)(du/dx)
Common Derivatives: d/dx(sin x) = cos x, d/dx(cos x) = -sin x, d/dx(e^x) = e^x, d/dx(ln x) = 1/x
Example: Differentiate y = x² e^x. Solution: dy/dx = e^x(2x + x²).
Applications of Derivatives
Increasing/Decreasing: f'(x) > 0 → increasing, f'(x) < 0 → decreasing.
Maxima/Minima: f'(x) = 0 → critical point. If f''(x) > 0 → local minimum, f''(x) < 0 → local maximum.
Example: f(x) = x² - 4x + 3. f'(x) = 2x - 4 = 0 → x=2. f''(x)=2>0 → Minimum at x=2, f(2)=-1.
Integrals
Definition: Integration is reverse of differentiation.
Standard Results: ∫ x^n dx = x^(n+1)/(n+1) + C (n≠-1) ∫ (1/x) dx = ln|x| + C ∫ e^x dx = e^x + C ∫ sin x dx = -cos x + C
Methods: 1. Substitution 2. By Parts: ∫ u v dx = u∫ v dx - ∫(du/dx ∫ v dx) dx 3. Partial Fractions
Example: ∫ x sin x dx = -x cos x + sin x + C.
Definite Integrals
