Intuitive Idea of Limits
A limit describes the value a function approaches as the input approaches some value.
We write lim_(x->a) f(x) = L to mean: as x gets arbitrarily close to a (but not equal to a), f(x) gets arbitrarily close to L.
Left-Hand and Right-Hand Limits
- Left-hand limit:
lim_(x->a^-) f(x)(approach from left, i.e.,x < a) - Right-hand limit:
lim_(x->a^+) f(x)(approach from right, i.e.,x > a)
A limit exists if and only if both left-hand and right-hand limits exist and are equal.
Algebra of Limits
If lim_(x->a) f(x) = L and lim_(x->a) g(x) = M:
lim_(x->a) [f(x) + g(x)] = L + Mlim_(x->a) [f(x) - g(x)] = L - Mlim_(x->a) [f(x) * g(x)] = L * Mlim_(x->a) [f(x)/g(x)] = L/M, providedM != 0lim_(x->a) [c * f(x)] = cLfor any constant c.
Limits of Polynomial Functions
lim_(x->a) (a_n x^n + a_(n-1) x^(n-1) + ... + a_0) = a_n a^n + a_(n-1) a^(n-1) + ... + a_0
For rational functions, substitute x = a if denominator != 0. If it gives 0/0, factorise or rationalise.
Limits of Trigonometric Functions
Fundamental limits:
lim_(x->0) sin x/x = 1(x in radians)lim_(x->0) tan x/x = 1lim_(x->0) (1 - cos x)/x = 0lim_(x->0) (1 - cos x)/x^2 = 1/2
Standard Limit Formulas
lim_(x->a) (x^n - a^n)/(x - a) = n a^(n-1)lim_(x->0) (e^x - 1)/x = 1lim_(x->0) (log(1+x))/x = 1lim_(x->0) (a^x - 1)/x = log a
Definition of Derivative
The derivative of f(x) at x = a is defined as:
f'(a) = lim_(h->0) (f(a+h) - f(a))/h (provided the limit exists).
Derivative as a Function
f'(x) = lim_(h->0) (f(x+h) - f(x))/h
Notation
- Leibniz:
dy/dx,df/dx - Lagrange:
f'(x) - Newton:
dot(y)
Differentiation by First Principle
Example: Find the derivative of f(x) = x^2.
f'(x) = lim_(h->0) ((x+h)^2 - x^2)/h
= lim_(h->0) (x^2 + 2xh + h^2 - x^2)/h
= lim_(h->0) (2xh + h^2)/h
= lim_(h->0) (2x + h) = 2x
Derivatives of Standard Functions
| Function | Derivative |
|---|---|
x^n | n x^(n-1) |
sin x | cos x |
cos x | -sin x |
tan x | sec^2 x |
e^x | e^x |
log x | 1/x (for x > 0) |
constant c | 0 |
Differentiation Rules
Sum Rule
d/dx [f(x) + g(x)] = f'(x) + g'(x)
Difference Rule
d/dx [f(x) - g(x)] = f'(x) - g'(x)
Product Rule
d/dx [f(x) * g(x)] = f'(x) * g(x) + f(x) * g'(x)
Memory aid: 'First times derivative of second + second times derivative of first.'
Quotient Rule
d/dx [f(x)/g(x)] = (f'(x)g(x) - f(x)g'(x))/(g(x))^2
Memory aid: 'Low dee-high minus high dee-low over low squared.'
Chain Rule
If y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).
Also written: dy/dx = dy/du * du/dx where u = g(x).
Worked Examples
Example 1: Find lim_(x->2) (x^2 - 4)/(x - 2).
Solution: Direct substitution gives 0/0. Factorise numerator: (x-2)(x+2)/(x-2) = x+2. Limit = 2 + 2 = 4.
Example 2: Find lim_(x->0) sin 3x/x.
Solution: = lim_(x->0) 3 * sin 3x/(3x) = 3 * 1 = 3.
Example 3: Differentiate f(x) = x^3 sin x using product rule.
Solution: f'(x) = 3x^2 * sin x + x^3 * cos x = x^2(3 sin x + x cos x).
Example 4: Differentiate f(x) = sin(x^2) using chain rule.
Solution: f'(x) = cos(x^2) * 2x = 2x cos(x^2).
Common Mistakes
- 0/0 indeterminate form: Do not conclude limit does not exist when
0/0. Simplify first. - Left and right limits: For piecewise functions, always check both.
- Product rule: Never differentiate term-by-term like
d/dx(x * x) = 1 * 1 = 1. Correct is1*x + x*1 = 2x. - Chain rule depth: When functions are nested, differentiate from outside in, one layer at a time.
ISC Exam Focus
- Theory (70%): Limit evaluation, first principle derivation, standard derivatives and rules.
- Application (30%): Numerical problems on limits, product/quotient/chain rules.
- ISC emphasis on
lim_(x->0) sin x/x = 1andlim_(x->a) (x^n - a^n)/(x-a) = na^(n-1). - Differentiation by first principle is a must-practice topic.
Self-Test Questions
Q1: Find lim_(x->1) (x^3 - 1)/(x - 1).
Answer: Using (x^n - a^n)/(x-a) formula: n=3, a=1. Limit = 3*1^2 = 3. Or factor: (x-1)(x^2+x+1)/(x-1) = x^2+x+1. At x=1: 1+1+1=3.
Q2: Find lim_(x->0) (sin 5x)/(sin 2x).
Answer: = lim_(x->0) (sin 5x/(5x) * 5x)/(sin 2x/(2x) * 2x) = (1*5x)/(1*2x) = 5/2.
Q3: Differentiate f(x) = 3x^2 + 2x - 5 from first principle.
Answer: f'(x) = lim_(h->0) [3(x+h)^2 + 2(x+h) - 5 - (3x^2+2x-5)]/h = lim_(h->0) (6xh + 3h^2 + 2h)/h = 6x + 2.
Q4: Differentiate y = e^x cos x.
Answer: dy/dx = e^x cos x + e^x(-sin x) = e^x(cos x - sin x).
Q5: Differentiate y = (x^2 + 1)/(x - 1).
Answer: Using quotient rule: dy/dx = ((2x)(x-1) - (x^2+1)(1))/(x-1)^2 = (2x^2 - 2x - x^2 - 1)/(x-1)^2 = (x^2 - 2x - 1)/(x-1)^2.
Q6: Differentiate y = e^(sin x).
Answer: Using chain rule: dy/dx = e^(sin x) * cos x.
