Holes
Example 1

Find the points of discontinuity for the rational function.

frac1

Step 1. Factor the rational function completely.

frac2

Step 2. Determine the points of discontinuity by setting the factors in the denominator equal to zero and solving.

equation1 or equation2

Unequal1 or Unequal2

Step 3. Simplify the rational function.

cancel

frac3

Step 4. Determine the vertical asymptote.

x = 5 is a vertical asymptote because it is a zero of the denominator only.


Step 5. Determine the coordinates of the hole.

There is a hole at x = -5 because the factor (x + 5) was cancelled in both the numerator and denominator. To determine the y-value, plug x = -5 into the simplified function and solve.

frac4

Coordinates of the hole: Solution

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