- Topic
- Continuity
- Difficulty
- Standard
- Suggested time
- 3 min
- Marks
- 2
Continuity practice question 13
Classify the discontinuities of the function below.
Which make the denominator zero? At each one, is the numerator zero too, or not?
Factor both numerator and denominator. A factor that cancels marks a hole; a factor that survives in the denominator marks a vertical asymptote.
After cancelling, take the limit at each excluded point. A finite limit means removable; a limit of means infinite.
Theory UBC Math 100 §2.3CLP-1 §1.6Key results
Worked solution
Strategy
Factor and cancel. Where a factor cancels, the limit exists and the discontinuity is removable; where the denominator still vanishes after cancelling, the function blows up.
Working
The factor (x − 2) cancels.
The limit exists at 2 although h(2) is undefined: a hole, which is removable.
At −2 the denominator still vanishes while the numerator is −1: an infinite discontinuity.
Answer
The answer is a removable discontinuity at 2 and an infinite discontinuity at −2.
Check
h(2.01) ≈ 0.75 stays near 3/4, while h(−1.99) = −99 and h(−2.01) = 101 blow up.
Where this goes wrong
Calling both points infinite discontinuities because the original denominator vanishes at both. Whether the function blows up depends on what is left after cancelling.
Leans on
Factoring quadratics, from Functions, graphs and inverses
Try a similar question
Find and classify every discontinuity of the function below.Saved in this browser only.