One-Sided Limits

Definition

One-sided limits describe the behavior of a function as the input approaches a specific value $c$ from only one direction—either from the left ($x < c$) or from the right ($x > c$).

Why It Matters

One-sided limits are the “Boundary Patrol” of calculus. They tell us what happens at the “edge” of a function’s domain. In real-world engineering, they are essential for modeling “step functions”—like a light switch turning on or a computer bit flipping from 0 to 1. If the left-hand and right-hand limits don’t agree, you have a “Jump Discontinuity”—a signal that the system has undergone a fundamental state change. Mastering these limits is how we ensure that our digital models of the continuous world remain accurate at the points of transition.

Core Concepts

• Right-Hand Limit: $\lim_{x \to c^+} f(x) = L$. The value $f(x)$ approaches $L$ as $x$ approaches $c$ from the right ($x > c$).
  • How to read: “The limit as x approaches c from the right of the function f of x is equal to L.”
  • Meaning: Describes behavior just to the right of $c$; essential at endpoints and jump discontinuities.
• Left-Hand Limit: $\lim_{x \to c^-} f(x) = L$. The value $f(x)$ approaches $L$ as $x$ approaches $c$ from the left ($x < c$).
  • How to read: “The limit as x approaches c from the left of the function f of x is equal to L.”
  • Meaning: Describes behavior just to the left of $c$; pairs with the right-hand limit to test continuity.
• Existence of Two-Sided Limit: $\lim_{x \to c} f(x) = L$ if and only if both one-sided limits exist and are equal: $\lim_{x \to c^-} f(x) = L \text{ and } \lim_{x \to c^+} f(x) = L$
  • How to read: “The two-sided limit is equal to L if and only if both the left-hand limit and the right-hand limit are equal to L.”
  • Meaning / when to use: If the one-sided limits disagree, the ordinary limit does not exist—typical at piecewise boundaries.

Connected Concepts

• One-Sided Derivatives
• One-to-One Functions
• Inverse Functions
• Fundamental Trigonometric Limits