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Cogito

AP Calculus AB · Chapter 1 · Lesson 3

Practice Worksheet

Computing Limits and Indeterminate Forms

What to do when substitution fails.

12 problems · about 22 minutes · LIM-1.D, LIM-1.E

NameDate

Figure — use these to answer the problems

Slide the point toward a value and watch the height settle. That settling is what a limit measures.
-8-8-6-6-4-4-2-222446688
y = 1x² + 1x + 0
  • Point(1, 2)
a1
b1
Point on the curve1
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. Limit of (x² − 1) ÷ (x − 1) as x approaches 1?

    Answer

  2. What does the Squeeze Theorem require?

    • The function is trapped between two others sharing a limit.
    • The function is continuous.

  3. Limit of (x² − 4) ÷ (x − 2) as x approaches 2?

    Answer

  4. Limit of (x² − 25) ÷ (x − 5) as x approaches 5?

    Answer

  5. Limit of (x³ − 8) ÷ (x − 2) as x approaches 2?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. Limit of (1 − cos x) ÷ x as x approaches 0?

    Answer

  2. Limit of 5x + 2 as x approaches 3?

    Answer

  3. g is trapped between −x² and x² near 0. What is its limit there?

    Answer

Stretch Yourself

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

4 problems

  1. Sort each form by what it tells you.

    Write each item under the heading it belongs to: Zero over zero · Infinity minus infinity · Five over zero · Negative three over zero

    More work needed

    Limit does not exist

  2. Limit of (x² − 16) ÷ (x + 4) as x approaches −4?

    Answer

  3. The Cancel

    Limit of (x² − 9) ÷ (x − 3) as x approaches 3?

    Answer

  4. The Special Limit

    Limit of sin x ÷ x as x approaches 0?

    Answer