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Cogito

Calculus · Chapter 2 · Lesson 2

Practice Worksheet

Types of Discontinuity

Three ways a graph can break.

12 problems · about 21 minutes · LIM-2.A, LIM-2.B

NameDate

Figure — use these to answer the problems

This curve has an infinite discontinuity at zero. The values do not settle; they run away.
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y = 1/x + 0
  • Point(2, 0.50)
a1
Point on the curve2
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. Left limit 3, right limit 3, function value 8. Which kind? 1, 2 or 3.

    Answer

  2. Why is a removable discontinuity called removable?

    • Redefining the function at that one point closes the hole.
    • Because the gap is small.

  3. Left limit 4, right limit 4, function undefined. Which kind? 1, 2 or 3.

    Answer

  4. Left limit 1, right limit 6. Which kind? 1, 2 or 3.

    Answer

  5. Values run away near the point. Which kind? 1, 2 or 3.

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. f(x) = (x² − 25) ÷ (x − 5). What repairs it at x = 5?

    Answer

  2. f(x) = 1 ÷ (x − 4). Which kind at x = 4? 1, 2 or 3.

    Answer

  3. How many kinds of discontinuity are named here?

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Match each description with its kind of discontinuity.

    Draw a line from each item on the left to its match on the right.

    • A single hole in the graph
    • The graph steps to a new height
    • The graph runs off the screen
    • Removable
    • Jump
    • Infinite

  2. Which kind can be repaired by redefining one point? 1, 2 or 3.

    Answer

  3. The Hole

    f(x) = (x² − 9) ÷ (x − 3). What value at x = 3 would repair the break?

    Answer

  4. The Step

    Left limit 2, right limit 5. Which kind of discontinuity? Enter 1 removable, 2 jump, 3 infinite.

    Answer