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).
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