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Infinite Discontinuity — Definition, Formula & Examples

An infinite discontinuity is a point where a function's value increases or decreases without bound, typically producing a vertical asymptote on the graph. At least one of the one-sided limits at that point equals +∞+\infty or −∞-\infty.

A function ff has an infinite discontinuity at x=cx = c if ff is defined on a punctured neighborhood of cc and at least one of lim⁡x→c−f(x)\lim_{x \to c^-} f(x) or lim⁡x→c+f(x)\lim_{x \to c^+} f(x) is +∞+\infty or −∞-\infty. The function cannot be made continuous at cc by redefining f(c)f(c), making this a type of essential (nonremovable) discontinuity.

How It Works

To identify an infinite discontinuity, find the points where the denominator of a rational function equals zero while the numerator does not. At each such point, evaluate the left-hand and right-hand limits. If either one-sided limit diverges to ±∞\pm\infty, the function has an infinite discontinuity there. On a graph, this appears as the curve shooting upward or downward near a vertical asymptote.

Worked Example

Problem: Determine whether f(x) = 1/(x − 2) has an infinite discontinuity at x = 2.
Step 1: Evaluate the left-hand limit as x approaches 2.
lim⁡x→2−1x−2=−∞\lim_{x \to 2^-} \frac{1}{x - 2} = -\infty
Step 2: Evaluate the right-hand limit as x approaches 2.
lim⁡x→2+1x−2=+∞\lim_{x \to 2^+} \frac{1}{x - 2} = +\infty
Step 3: Since both one-sided limits diverge to infinity, the function has an infinite discontinuity at x = 2. The line x = 2 is a vertical asymptote.
Answer: Yes, f(x) = 1/(x − 2) has an infinite discontinuity at x = 2.

Why It Matters

Recognizing infinite discontinuities is essential in Calculus I when sketching curves, finding domains, and evaluating improper integrals. Engineers and physicists encounter them when modeling phenomena like gravitational fields or electric potentials near point sources, where quantities blow up at specific locations.

Common Mistakes

Mistake: Confusing an infinite discontinuity with a removable discontinuity when both the numerator and denominator are zero.
Correction: If both numerator and denominator are zero, factor and simplify first. A common factor that cancels indicates a removable discontinuity (a hole), not an infinite one. An infinite discontinuity only occurs when the denominator vanishes but the numerator does not (after simplification).