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Cogito

Calculus · Chapter 1 · Lesson 3

Practice Worksheet

Limits at Infinity and Asymptotes

What happens at the far ends of a graph.

12 problems · about 21 minutes · LIM-2.D

NameDate

Figure — use these to answer the problems

Follow the curve outward and it flattens toward zero. Follow it inward and it runs away.
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y = 2/x + 0
  • Point(3, 0.67)
a2
Point on the curve3
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. Limit of (8x + 3) ÷ (4x − 1) as x grows without bound?

    Answer

  2. A denominator is zero and the numerator is not. What is at that point?

    • A vertical asymptote.
    • A hole.

  3. Limit of (4x) ÷ (x + 1) as x grows without bound?

    Answer

  4. Limit of (x + 2) ÷ (x² + 1) as x grows without bound?

    Answer

  5. Limit of (9x² + x) ÷ (3x² − 4) as x grows without bound?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. f(x) = 5 ÷ (x + 2). Where is the vertical asymptote?

    Answer

  2. f(x) = 1 ÷ x. What is the horizontal asymptote value?

    Answer

  3. f(x) = (x² − 4) ÷ (x − 2). Is x = 2 an asymptote or a hole? 1 for asymptote, 2 for hole.

    Answer

Stretch Yourself

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

4 problems

  1. Which produce a horizontal asymptote of y = 0?

    • Degree 1 over degree 2
    • Degree 2 over degree 5
    • Degree 3 over degree 3
    • Degree 4 over degree 2
    • Tick every box that applies.

  2. Limit of (7x³) ÷ (2x³ + 9) as x grows without bound, as a decimal?

    Answer

  3. The Far End

    Limit of (6x + 1) ÷ (2x − 5) as x grows without bound?

    Answer

  4. The Break

    f(x) = 1 ÷ (x − 7). At which x is the vertical asymptote?

    Answer