Graphical Limits: Interactive Guide

The easiest way to think about limits is to imagine you're on a roller coaster riding the track from one side or the other:

Left-Hand Limit (x ➔ c⁻)

• What it means: Tracing the track coming from the left-hand side (indicated by the negative exponent superscript).

• In the simulation: Drag the red roller coaster cart from the left toward the target x-value (e.g., x = 5). As it gets closer, it turns blue, showing you the height (y-value) f(x) approaches from the left-hand side.

Formula

\lim_{x \to c^-} f(x) = L

Easy Example Problem

Trace a piecewise graph to determine the limit of f(x) as x approaches 2, where f(x) has a hole at (2, 3) and a point at (2, 1).

1. Tracing the curve from the left: as x approaches 2 from values less than 2, the y-value approaches 3.

2. Tracing the curve from the right: as x approaches 2 from values greater than 2, the y-value also approaches 3.

3. Because both one-sided limits match:

\lim_{x \to 2^{-}} f(x) = 3 \quad \text{and} \quad \lim_{x \to 2^{+}} f(x) = 3 \implies \lim_{x \to 2} f(x) = 3

Note that the actual value of the function f(2) = 1 does not affect the limit.

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