Types of Continuity and Discontinuity Math Example 3

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Example 3

easy
Classify the discontinuity of g(x)=1(x−3)2g(x) = \frac{1}{(x-3)^2} at x=3x = 3.

Solution

  1. 1
    g(3)g(3) is undefined.
  2. 2
    As x→3x \to 3, (x−3)2→0+(x-3)^2 \to 0^+, so g(x)→+∞g(x) \to +\infty.
  3. 3
    This is an infinite discontinuity (vertical asymptote at x=3x = 3).

Answer

Infinite discontinuity at x=3x = 3.
An infinite discontinuity occurs when the function blows up near a point. The vertical asymptote at x=3x = 3 is the geometric manifestation.

About Types of Continuity and Discontinuity

Continuity types classify how a function can fail to be continuous at a point. A removable discontinuity (hole) occurs when the limit exists but doesn't equal f(a). A jump discontinuity occurs when left and right limits differ. An infinite discontinuity occurs when the function approaches ±∞.

Learn more about Types of Continuity and Discontinuity →

More Types of Continuity and Discontinuity Examples

Example 1 easy

Classify the discontinuity of [formula] at [formula].

Example 2 medium

Determine whether the piecewise function [formula] is continuous at [formula].

Example 4 medium

Find the value of [formula] that makes [formula] continuous at [formula].

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