Second Kind Discontinuity

A discontinuity of the second kind is another important type of discontinuity; its characteristic feature is that at least one one-sided limit fails to exist.

Definition

Characteristics

• At least one one-sided limit fails to exist
• The function may be unbounded in a neighbourhood of that point
• The graph of the function may have a vertical asymptote at that point

Common Examples

Example 1: Rational Function

Function: $f(x) = \frac{1}{x}$ at $x = 0$

Analysis:

• The function is undefined at $x = 0$
• Left limit: $\lim_{x \to 0^-} \frac{1}{x} = -\infty$
• Right limit: $\lim_{x \to 0^+} \frac{1}{x} = +\infty$
• Neither the left nor the right limit exists (both tend to infinity)
• Therefore $x = 0$ is a discontinuity of the second kind

Graph Features:

• The graph of the function has a vertical asymptote at $x = 0$
• It tends to negative infinity on the left and to positive infinity on the right

Example 2: Reciprocal Square Function

Function: $f(x) = \frac{1}{x^2}$ at $x = 0$

Analysis:

• The function is undefined at $x = 0$
• Left limit: $\lim_{x \to 0^-} \frac{1}{x^2} = +\infty$
• Right limit: $\lim_{x \to 0^+} \frac{1}{x^2} = +\infty$
• Neither the left nor the right limit exists (both tend to positive infinity)
• Therefore $x = 0$ is a discontinuity of the second kind

Graph Features:

• The graph of the function has a vertical asymptote at $x = 0$
• It tends to positive infinity on both sides

Example 3: Trigonometric Function

Function: $f(x) = \tan x$ at $x = \frac{\pi}{2}$

Analysis:

• The function is undefined at $x = \frac{\pi}{2}$
• Left limit: $\lim_{x \to (\frac{\pi}{2})^-} \tan x = +\infty$
• Right limit: $\lim_{x \to (\frac{\pi}{2})^+} \tan x = -\infty$
• Neither one-sided limit exists (they tend to infinity with opposite signs)
• Therefore $x = \frac{\pi}{2}$ is a discontinuity of the second kind

Graph Features:

• The graph of the function has a vertical asymptote at $x = \frac{\pi}{2}$
• It tends to positive infinity on the left and to negative infinity on the right

Example 4: Exponential Function

Function: $f(x) = e^{\frac{1}{x}}$ at $x = 0$

Analysis:

• The function is undefined at $x = 0$
• Left limit: $\lim_{x \to 0^-} e^{\frac{1}{x}} = 0$ (because $\frac{1}{x} \to -\infty$)
• Right limit: $\lim_{x \to 0^+} e^{\frac{1}{x}} = +\infty$ (because $\frac{1}{x} \to +\infty$)
• The right limit does not exist
• Therefore $x = 0$ is a discontinuity of the second kind

Graph Features:

• It tends to 0 on the left
• It tends to positive infinity on the right

How to Handle Them

A discontinuity of the second kind usually indicates a severe discontinuity of the function in a neighbourhood of that point:

Analyse the Behaviour of the Function

• Study the behaviour of the function near that point
• Analyse whether it affects the overall properties of the function
• Determine the position of the vertical asymptote

Exercises

Exercise 1

Determine the type of discontinuity of $f(x) = \frac{1}{x^2}$ at $x = 0$.

Exercise 2

Determine the type of discontinuity of $f(x) = \tan x$ at $x = \frac{\pi}{2}$.

Exercise 3

Determine the type of discontinuity of $f(x) = \frac{1}{x-1}$ at $x = 1$.

Exercise 4

Determine the type of discontinuity of $f(x) = e^{\frac{1}{x}}$ at $x = 0$.

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Summary

Symbols Used in This Article