Limits and Derivatives

Calculus begins with the concept of a limit. Informally, the limit of a function f(x) as x approaches a value c is the value that f(x) gets closer and closer to (but may not actually reach) as x approaches c. This chapter introduces limits and the derivative as a rate of change.

Limits:
We write: lim(x→c) f(x) = L to mean f(x) → L as x → c.

Standard Limits:
lim(x→0) sin(x)/x = 1
lim(x→0) (1 - cos x)/x = 0
lim(x→a) (xⁿ - aⁿ)/(x - a) = n × a^n-1
lim(x→0) (e^x - 1)/x = 1
lim(x→0) log(1 + x)/x = 1

Algebra of Limits:
If lim f(x) = L and lim g(x) = M, then:
lim [f(x) + g(x)] = L + M
lim [f(x) × g(x)] = L × M
lim [f(x)/g(x)] = L/M (provided M ≠ 0)

Derivatives:
The derivative of f(x) at x = a measures the instantaneous rate of change. It is defined as:
f'(a) = lim(h→0) [f(a+h) - f(a)] / h

This is called differentiation from first principles.

Standard Derivatives:
d/dx (xⁿ) = nx^n-1
d/dx (sin x) = cos x
d/dx (cos x) = -sin x
d/dx (tan x) = sec² x
d/dx (constant) = 0

Rules of Differentiation:
Sum/Difference Rule: (f ± g)' = f' ± g'
Product Rule: (fg)' = f'g + fg'
Quotient Rule: (f/g)' = (f'g - fg')/g²

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Worked Examples

Example 1

Evaluate lim(x→2) (x² - 4)/(x - 2).
Factor: (x² - 4)/(x - 2) = (x+2)(x-2)/(x-2) = x + 2. As x→2, limit = 4

Example 2

Evaluate lim(x→0) sin(3x)/x.
= lim(x→0) 3 × sin(3x)/(3x) = 3 × 1 = 3

Example 3

Find lim(x→1) (x³ - 1)/(x - 1).
Using formula: lim(x→a) (xⁿ - aⁿ)/(x-a) = na^n-1. Here n = 3, a = 1. Limit = 3 × 1² = 3

Example 4

Differentiate f(x) = 3x⁴ - 2x² + 5 from first principles OR using rules.
f'(x) = 12x³ - 4x + 0 = 12x³ - 4x

Example 5

Find d/dx (x² sin x).
Using product rule: d/dx = 2x × sin x + x² × cos x = 2x sin x + x² cos x

Example 6

Differentiate y = (x² + 1)/(x - 1) using the quotient rule.
dy/dx = [(2x)(x-1) - (x²+1)(1)] / (x-1)² = [2x² - 2x - x² - 1] / (x-1)² = (x² - 2x - 1)/(x-1)²

Example 7

Find the derivative of f(x) = x³ - 2x at x = 1 from first principles.
f'(1) = lim(h→0) [(1+h)³ - 2(1+h) - (1-2)]/h
= lim(h→0) [1 + 3h + 3h² + h³ - 2 - 2h + 1]/h
= lim(h→0) [h + 3h² + h³]/h = lim(h→0) (1 + 3h + h²) = 1

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Common mistakes

Key formulas

You cannot simply substitute x = c in a limit if it gives 0/0. Factor the expression first.

The product rule is (fg)' = f'g + fg', NOT (fg)' = f'g'. Many students forget the second term.

lim sin(x)/x = 1 only when x is in radians and the argument of both sin and denominator is the same.

Summary

Limits describe the behaviour of functions near a point. Derivatives measure instantaneous rates of change. Master the standard formulas and differentiation rules — these are the gateway to all of calculus.

Worked Examples

Practice Problems