Calculus — Functions, Limits, and Continuity

What Is Calculus?

Calculus is the branch of mathematics that deals with change and accumulation. Developed independently in the 17th century by Newton and Leibniz, it is the language of physics, engineering, economics, biology, and virtually every science.

The two pillars of calculus:

1. Differential calculus: computing instantaneous rates of change
2. Integral calculus: computing area and accumulated quantities

The relationship between differentiation and integration:

• The Fundamental Theorem of Calculus: differentiation and integration are inverse operations
• The Newton-Leibniz theorem

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A Review of Functions

The definition of a function f: A → B:
→ A rule that assigns exactly one element f(x) of set B
  to every element x of set A

Domain: the set of possible inputs x
Codomain: the set of possible outputs y
Range: the set of actual output values y

Examples of functions:
→ f(x) = x²: domain R, range [0, ∞)
→ f(x) = 1/x: domain R\{0} (excluding 0)
→ f(x) = √x: domain [0, ∞)

Types of functions:
→ Polynomial functions: f(x) = aₙxⁿ + ... + a₀
→ Rational functions: a ratio of polynomials (denominator ≠ 0)
→ Trigonometric functions: sin, cos, tan, etc.
→ Exponential functions: f(x) = aˣ (a > 0, a ≠ 1)
→ Logarithmic functions: f(x) = logₐ(x)

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An Intuitive Understanding of the Limit

Intuitive limit:
→ The value f(x) approaches as x approaches a

Notation: lim f(x) = L
           x→a

Example:
→ lim (x² − 1)/(x − 1) = ?
   x→1

Direct substitution: at x=1 the denominator = 0 → can't evaluate directly

Factoring:
→ (x² − 1)/(x − 1) = (x−1)(x+1)/(x−1) = x+1 (x≠1)
→ As x approaches 1: x+1 → 2
→ So the limit = 2

The key point:
→ A limit is not the function's value at x=a,
  it's the value f(x) approaches as x nears a

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Left-Hand and Right-Hand Limits

Right-hand limit:
→ As x approaches a from the right (larger side)
→ lim f(x) = L⁺
   x→a⁺

Left-hand limit:
→ As x approaches a from the left (smaller side)
→ lim f(x) = L⁻
   x→a⁻

The condition for a limit to exist:
→ If the left-hand limit = the right-hand limit = L, then
  lim f(x) = L exists
  x→a

Example: the absolute-value function f(x) = |x| / x (x ≠ 0)
→ x→0⁺: f(x) → +1
→ x→0⁻: f(x) → −1
→ Left-hand limit ≠ right-hand limit → the limit does not exist

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Properties of Limits

Limit laws (if lim f(x) = L and lim g(x) = M):
                 x→a             x→a

1. Sum: lim [f(x) + g(x)] = L + M
2. Difference: lim [f(x) − g(x)] = L − M
3. Product: lim [f(x) · g(x)] = L · M
4. Quotient: lim [f(x) / g(x)] = L/M (provided M ≠ 0)
5. Constant multiple: lim [c · f(x)] = c · L
6. Power: lim [f(x)]ⁿ = Lⁿ

The Squeeze Theorem:
→ If g(x) ≤ f(x) ≤ h(x) and
  lim g(x) = lim h(x) = L, then
  lim f(x) = L

Application:
→ lim x·sin(1/x) = 0 (x→0)
   (since |sin| ≤ 1)

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Limits at Infinity

Limits as x → ∞:

• The behavior of f(x) as x grows without bound

Example:

• lim (3x² + 2x)/(x² − 1) = ? x→∞

Divide numerator and denominator by x²:

• lim (3 + 2/x)/(1 − 1/x²) = 3/1 = 3 x→∞

Horizontal asymptote:

• If lim f(x) = L, then y = L is a horizontal asymptote x→±∞

Vertical asymptote:

• If lim f(x) = ±∞, then x = a is a vertical asymptote x→a

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Continuity

The three conditions for continuity at x = a:

1. f(a) is defined
2. lim f(x) exists x→a
3. lim f(x) = f(a) x→a

• All three conditions must hold for continuity

Types of discontinuity:

1. Removable discontinuity: the limit exists but f(a) ≠ the limit
2. Jump discontinuity: the left-hand limit ≠ the right-hand limit
3. Infinite discontinuity: the limit is ±∞

Properties of continuous functions:

• Polynomial, trigonometric, exponential, and logarithmic functions: continuous on their domain
• Sums, differences, products, and compositions of continuous functions: continuous

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The Intermediate Value Theorem

The Intermediate Value Theorem:

• If f is continuous on [a, b] and f(a) ≠ f(b), then for every value k between f(a) and f(b), there exists a c ∈ (a, b) such that f(c) = k

Intuition:

• A continuous function cannot “skip over” a value
• If you travel from a place at 0°C to a place at 30°C, you pass through every temperature in between

Application:

• Proving a root exists for an equation
• Example: f(x) = x³ − x − 2 f(1) = −2 < 0, f(2) = 4 > 0
  • by the Intermediate Value Theorem, a root exists in (1, 2)

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Frequently Asked Questions

Q. Can a function’s limit differ from its actual value? A. Yes, easily. A function’s limit can exist even where the function is not defined at x=a. For example, f(x) = (x²−1)/(x−1) is undefined at x=1, but its limit there is 2. Conversely, a function can be defined at a point yet still have a limit that differs from its function value there.

Q. Do I really need to know the epsilon-delta definition? A. For most engineering and economics majors, the intuitive concept of a limit is sufficient. But if you’re a math major, or need mathematical rigor, you should understand the epsilon-delta definition. We’ll work with it in more depth in Chapter 2, when we cover derivatives.