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Discontinuity Classifier

Classify discontinuities of rational functions by analyzing limits

Rational function: f(x) = (ax+b)/(cx+d). Check at x = p

f(x) = (x+)/(x+), analyze at x =

Discontinuity Types

Removable (hole): limit exists finite, f(a) undefined

Jump: left and right limits are finite but differ

Infinite: limit goes to +/-infinity (asymptote)

To classify: check f(a), lim left, lim right

Discontinuities occur where a function is not continuous. For rational functions, denominator zeros cause discontinuities. The type depends on whether the factor cancels with the numerator (hole) or not (asymptote).

⚠ A discontinuity is classified by analyzing limits. Removable = limit exists (finite). Jump = limits differ. Infinite = limit is infinite.

What Is a Discontinuity?

A discontinuity is a point where a function is not continuous. For rational functions, denominator zeros always create discontinuities. If the numerator also has the same zero, the factor cancels and creates a hole (removable). Otherwise it creates an asymptote (infinite).

Removable (Hole)

Limit exists, but f(a) undefined. Factor cancels. Fix by redefining f(a)=limit. (x^2-4)/(x-2) has hole at x=2.

Infinite (Asymptote)

Function goes to +/-inf. Denominator zero, numerator non-zero. f(x)=1/(x-2) has VA at x=2.

Jump

Left and right limits are finite but unequal. Common in piecewise functions. Can be removed by adjusting the pieces.

Classification Method

Check denominator=0 at a. If numerator also 0: check if factor cancels (hole). Else: asymptote. Compute limits to confirm.

Teaching Example: f(x)=(x-4)/(x-2) at x=2. Denominator=0, numerator=2-4=-2 != 0. f undefined. Left limit -> +inf, right limit -> -inf. INFINITE discontinuity (VA).

Applications

Calculus Graphing Limit Analysis Rational Functions Continuity

Frequently Asked Questions

What are discontinuity types?▼

Removable (hole): limit exists. Jump: sides differ. Infinite: goes to inf. Classify by analyzing limits.

How to classify?▼

Check if denominator=0. Test numerator. Compute left and right limits. Compare results.

Hole vs asymptote?▼

Hole: factor cancels (numerator also zero at same point). Asymptote: denominator zero, numerator non-zero.

Can a discontinuity be removed?▼

Only removable discontinuities (holes) can be fixed by redefining f at the point. Asymptotes and jumps are permanent.

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