Section 2.3

One-Sided Limits

When functions behave differently depending on the direction of approach—left or right.

Theory: Direction Matters

Right-Hand Limit

Approaching from values greater than

Left-Hand Limit

Approaching from values less than

Theorem of Existence

The general limit exists if and only if both one-sided limits exist and are equal to .

Application: The Heaviside Step Function

The Heaviside step function is defined as:

Left-Hand Limit

Right-Hand Limit

Since , the limit at does not exist.

Engineering Context: Signal Processing

This function is fundamental in electrical engineering and control theory. It models the closing of a switch in a circuit.

Analog Signals

Continuous functions, like sound waves on vinyl. Limits and continuity describe them.

Digital Signals

Discrete steps, like binary code. Fundamentally piecewise with jump discontinuities.

Example: Absolute Value Function

Consider :

For

, so

Limit from right:

For

, so

Limit from left:

This is the signum (sign) function, critical in computing for extracting the sign of a number.

Example: The Floor Function

The greatest integer function returns the largest integer less than or equal to .

At any integer :

Real-World Application

This models step-costs, such as shipping rates that jump at specific weight thresholds, or postage stamp pricing.

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Interactive Exploration

Explore three classic examples of one-sided limits. Click the tabs to see each function's jump discontinuity.

Heaviside H(t)

Step function: jumps from 0 to 1 at t=0

Left limit = 0, Right limit = 1 → Limit at t=0 does NOT exist