Infinite Discontinuity

Core Definition

A function $f(x)$ has an infinite discontinuity at $x = c$ if one or both of the one-sided limits grow without bound towards positive or negative infinity:

\lim_{x \to c^-} f(x) = \pm\infty \quad \text{or} \quad \lim_{x \to c^+} f(x) = \pm\infty

Geometrically, this means the line

x = c

is a vertical asymptote for the graph of

f(x)

Concept Explanation

At an infinite discontinuity, the function exhibits unbounded behavior. As $x$ pinches closer to $c$ , the output values shoot infinitely high ( $+\infty$ ) or infinitely deep ( $-\infty$ ).

        y ^        | (Vertical Asymptote x = c)
          |        |
          |  |     |     |
          |  |     |     |
          |  \     |     /
          |   \    |    /
          +----+---+---+----------> x
               |   c   |

Because infinity is not a real number, any limit yielding $\pm\infty$ fails to exist in the real numbers ( $\text{DNE}$ ).

Algebraic Hallmarks: $\frac{k}{0}$ vs $\frac{0}{0}$

When evaluating limits of rational functions by direct substitution:

$\frac{0}{0}$ (Indeterminate): Indicates a Removable Discontinuity (hole) that can be factored and canceled.
$\frac{k}{0}$ with $k \neq 0$ (Undefined / Asymptote): Indicates an Infinite Discontinuity (vertical asymptote). The expression cannot be canceled; you must analyze the sign using Strategies for One-Sided Limits (the "positive and negative zero" approach).

Example Problems

Example Problem 1: Two-Sided Asymptote Analysis

Evaluate the one-sided and two-sided limits for:

f(x) = \frac{1}{x - 3}

Solution

Step 1: Left-hand limit ( $x \to 3^-$ )
As $x \to 3$ from the left ( $x < 3$ ), $x - 3$ is a tiny negative decimal ( $0^-$ ):

\lim_{x \to 3^-} \frac{1}{x - 3} = \frac{1}{0^-} = -\infty

Step 2: Right-hand limit ( $x \to 3^+$ )
As $x \to 3$ from the right ( $x > 3$ ), $x - 3$ is a tiny positive decimal ( $0^+$ ):

\lim_{x \to 3^+} \frac{1}{x - 3} = \frac{1}{0^+} = +\infty

Step 3: Two-sided limit and classification
Since the one-sided limits blow up to opposite infinities, $\lim_{x \to 3} f(x) = \text{DNE}$ . $f(x)$ has an infinite discontinuity with vertical asymptote $x = 3$ .

Example Problem 2: Even Power Vertical Asymptote

Evaluate the limit:

\lim_{x \to 2} \frac{5}{(x - 2)^2}

Solution

Step 1: Check denominator behavior from both sides
As $x \to 2$ from either side ( $x \to 2^-$ or $x \to 2^+$ ), $(x - 2)^2$ is squared, so it is strictly positive and approaches $0$ ( $0^+$ ).

Step 2: Evaluate the limit

\lim_{x \to 2} \frac{5}{(x - 2)^2} = \frac{5}{0^+} = +\infty

Conclusion: $f(x)$ has an infinite discontinuity at $x = 2$ , where both sides shoot upward to $+\infty$ .

Connections & References

Parent Concepts: Discontinuities, Continuity
Related Concepts: Vertical Asymptotes, Strategies for One-Sided Limits, Existence Theorem for Two-Sided Limits
Course Links: Calc-1 Session 02 | Calc-1 Session 04