Elementary Analysis

The Intuitive Notion of Limit

We write $\lim_{x \to x_0} f(x) = K$ if $f(x)$ can be made arbitrarily close to $K$ by taking $x$ sufficiently close to $x_0$ .

Key point: This is about behaviour near $x_0$ , not at $x_0$ . The limit can exist even if $f(x_0)$ is undefined or different from $K$ .

The (ε, δ) Definition (Weierstrass)

$\lim_{x \to x_0} f(x) = K$ if and only if: for any $\varepsilon > 0$ , there exists $\delta > 0$ such that $|f(x) - K| < \varepsilon$ for all $x$ satisfying $0 < |x - x_0| < \delta$ .

$Function graph showing x₀, with ε-band around K and corresponding δ-interval around x₀, illustrating the ε-δ definition$

Key features:

The order matters: for any $\varepsilon$ , we must find some $\delta$
The condition $0 < |x - x_0|$ excludes $x_0$ itself
We control $x$ (through $\delta$ ) to control $f(x)$ (through $\varepsilon$ )

Analogy: A game where an opponent picks $\varepsilon$ , and you must respond with a valid $\delta$ .

Example: Proving $\lim_{x \to 0} \frac{\sin x}{x} = 1$

We need: for any $\varepsilon > 0$ , find $\delta$ such that $|x| < \delta \Rightarrow \left|\frac{\sin x}{x} - 1\right| < \varepsilon$ .

Using the inequality $|\sin x| < |x| < |\tan x|$ for $0 < x < \pi/2$ :

From $\sin x < x < \tan x$ , we get $\cos x < \frac{\sin x}{x} < 1$
As $x \to 0$ , $\cos x \to 1$ , so $\frac{\sin x}{x}$ is squeezed to 1

This uses the squeeze theorem: If $g(x) \leq f(x) \leq h(x)$ and $\lim g = \lim h = L$ , then $\lim f = L$ .

One-Sided Limits

Right-hand limit: $\lim_{x \to a^+} f(x)$

Left-hand limit: $\lim_{x \to a^-} f(x)$

The (two-sided) limit exists if and only if both one-sided limits exist and are equal.

Example: The Heaviside step function: $H(x) = \begin{cases} 0 & x < 0 \\ 1 & x \geq 0 \end{cases}$

has $\lim_{x \to 0^+} H(x) = 1$ and $\lim_{x \to 0^-} H(x) = 0$ . Since these differ, $\lim_{x \to 0} H(x)$ does not exist.

Limits at Infinity

$\lim_{x \to \infty} f(x) = K$ if for any $\varepsilon > 0$ , there exists $X$ such that $|f(x) - K| < \varepsilon$ for all $x > X$ .

Example: $\lim_{x \to \infty} \frac{1}{x} = 0$

Given $\varepsilon > 0$ , choose $X = 1/\varepsilon$ . Then for $x > X$ : $\left|\frac{1}{x} - 0\right| = \frac{1}{x} < \frac{1}{X} = \varepsilon$

Continuity

A function $f$ is continuous at $x = a$ if:

$f(a)$ exists (the function is defined there)
$\lim_{x \to a} f(x) = f(a)$

(ε, δ) form: For any $\varepsilon > 0$ , there exists $\delta > 0$ such that $|f(x) - f(a)| < \varepsilon$ for all $|x - a| < \delta$ .

Note: unlike the limit definition, this includes $x = a$ .

Types of discontinuity:

Removable: limit exists but ≠ function value
Jump: one-sided limits exist but differ
Essential: limit does not exist (e.g., oscillation)

Differentiability

A function $f$ is differentiable at $x = a$ if the limit:

$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

exists.

Key relationship:

Differentiable $\Rightarrow$ continuous
Continuous $\not\Rightarrow$ differentiable

Example where continuous but not differentiable: $f(x) = |x|$ at $x = 0$ . The left and right derivatives differ.

Big-O Notation

We write $f(x) = O(g(x))$ as $x \to a$ if there exist constants $K > 0$ and $\delta > 0$ such that:

$|f(x)| \leq K|g(x)| \quad \text{for all } 0 < |x - a| < \delta$

Intuition: $f$ is bounded by a multiple of $g$ near $a$ .

Examples:

$x^2 + x = O(x^2)$ as $x \to \infty$ (the $x^2$ term dominates)
$\sin x = O(x)$ as $x \to 0$ (since $\sin x / x \to 1$ )

Notation convention: We usually state the tightest bound.

Why Rigorous Analysis Matters

The (ε, δ) framework:

Makes precise what “close” means
Avoids circular reasoning in limits
Handles pathological cases (discontinuous functions, functions that oscillate)
Provides foundations for calculus proofs

Historically, calculus was developed before rigorous foundations. The (ε, δ) definition (Weierstrass, 19th century) resolved centuries of confusion about infinitesimals, convergence, and continuity.