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One-Sided Limit

One-Sided Limit

Either a limit from the left or a limit from the right.

Key Formula

lim⁡x→c−f(x)=Lorlim⁡x→c+f(x)=L\lim_{x \to c^-} f(x) = L \quad \text{or} \quad \lim_{x \to c^+} f(x) = L
Where:
  • c−c^- = Approaching c from the left (values less than c)
  • c+c^+ = Approaching c from the right (values greater than c)
  • LL = The value the function approaches from that side

Worked Example

Problem: Find the one-sided limits of the piecewise function f(x) = { 2x + 1 if x < 3, x² if x ≥ 3 } as x approaches 3.
Step 1: Find the limit from the left. As x approaches 3 from values less than 3, use the rule f(x) = 2x + 1.
lim⁡x→3−f(x)=2(3)+1=7\lim_{x \to 3^-} f(x) = 2(3) + 1 = 7
Step 2: Find the limit from the right. As x approaches 3 from values greater than 3, use the rule f(x) = x².
lim⁡x→3+f(x)=32=9\lim_{x \to 3^+} f(x) = 3^2 = 9
Step 3: Compare the two one-sided limits. Since 7 ≠ 9, the two-sided limit does not exist at x = 3.
lim⁡x→3−f(x)≠lim⁡x→3+f(x)\lim_{x \to 3^-} f(x) \neq \lim_{x \to 3^+} f(x)
Answer: The left-hand limit is 7 and the right-hand limit is 9. Because they differ, the ordinary (two-sided) limit at x = 3 does not exist.

Why It Matters

One-sided limits are essential for analyzing piecewise functions, jump discontinuities, and points where a function behaves differently on each side. The ordinary two-sided limit exists at a point only when both one-sided limits exist and are equal, so checking them is often the first step in determining whether a limit exists. They also appear in the formal definition of continuity and in evaluating limits at endpoints of a domain.

Common Mistakes

Mistake: Assuming that if both one-sided limits exist, the two-sided limit must also exist.
Correction: Both one-sided limits must exist AND be equal to each other for the two-sided limit to exist. If they have different values (as in the example above), the two-sided limit does not exist.

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