1.13 Removing Discontinuities

What is a removable discontinuity?

A removable discontinuity occurs at a point where the limit of a function exists but either the function is undefined there or its value doesn't match the limit. Graphically, this appears as a hole in the graph. It's called "removable" because you can fix it by redefining the function's value at that single point.

Finding the value that removes the hole

To remove a discontinuity at x = a, find lim(x→a) f(x). If this limit exists (both one-sided limits are equal and finite), define a new function g(x) equal to f(x) everywhere except at x = a, where g(a) = the limit value. This makes g continuous at a.

Common algebraic technique

Removable discontinuities often arise from a common factor in the numerator and denominator of a rational function that cancels out, like (x²-4)/(x-2) at x=2. Factor, simplify, then evaluate the limit by substitution. This reveals the value the function "should" have had, which becomes the redefined point.

Key terms

Removable discontinuity

A point where the limit exists but the function is undefined or doesn't equal the limit; appears as a hole in the graph.

How to remove a discontinuity

Redefine the function's value at that point to equal the limit, making it continuous there.

Common cause of removable discontinuities

A factor that cancels between the numerator and denominator of a rational function.

Condition for removability

The two-sided limit at that point must exist and be finite (one-sided limits are equal).

Example: f(x) = (x²-4)/(x-2)

Simplifies to x+2 for x≠2; limit as x→2 is 4, so defining f(2)=4 removes the discontinuity.

Try an AP-style question

f(x) = (x² + x − 12)/(x − 3). What value of f(3) removes the discontinuity?

1. 7
2. 3
3. −4
4. 0