Infinite limits and vertical asymptotes

About this lesson

Precalculus › Limits: an introduction

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Infinite limits and vertical asymptotes

Warm-up

Three quick questions from earlier lessons.

Discover

What happens to 1x\dfrac{1}{x} as xx approaches 0 from the right: 10.1,10.01,…\dfrac{1}{0.1}, \dfrac{1}{0.01}, \dots?

  1. It grows without bound
  2. It approaches 0

Key idea

Near a zero of the denominator, a fraction can blow up.

If f(x)f(x) grows without bound as x→a+x \to a^+ (or a−a^-), we write lim⁡f(x)=+∞\lim f(x) = +\infty (or −∞-\infty): the line x=ax = a is a vertical asymptote. Find the sign by testing a value just to the left or right of aa.

Worked example

Find lim⁡x→3−2x−3\displaystyle\lim_{x \to 3^-} \dfrac{2}{x - 3}.

x−3x - 3 is a small negative number
Just left of 3, for example x = 2.99.
The quotient is large and negative
2 divided by it.
lim⁡x→3−2x−3=−∞\displaystyle\lim_{x \to 3^-} \dfrac{2}{x - 3} = -\infty
Write the infinite limit.

Your turn, guided

Your turn: f(x)=x+1x2−16f(x) = \dfrac{x + 1}{x^2 - 16}. Where are the vertical asymptotes?

x=4x = 4 or x=−4x = -4
x² − 16 = 0.
Asymptotes x=4x = 4 and x=−4x = -4
The numerator is not 0 there. Type the negative one.

Practice ★

  1. What is lim⁡x→6+7x−6\displaystyle\lim_{x \to 6^+} \dfrac{ 7 }{x - 6}?
    1. +∞+\infty
    2. −∞-\infty
    3. 00
  2. What is lim⁡x→3+4x−3\displaystyle\lim_{x \to 3^+} \dfrac{ 4 }{x - 3}?
    1. +∞+\infty
    2. −∞-\infty
    3. 00
  3. What is lim⁡x→7+6x−7\displaystyle\lim_{x \to 7^+} \dfrac{ 6 }{x - 7}?
    1. +∞+\infty
    2. −∞-\infty
    3. 00

Watch out!

Practice ★★

  1. What is lim⁡x→8−6x−8\displaystyle\lim_{x \to 8^-} \dfrac{ 6 }{x - 8}?
    1. +∞+\infty
    2. −∞-\infty
    3. 00
  2. What is lim⁡x→2−7x−2\displaystyle\lim_{x \to 2^-} \dfrac{ 7 }{x - 2}?
    1. +∞+\infty
    2. −∞-\infty
    3. 00
  3. What is lim⁡x→5−9x−5\displaystyle\lim_{x \to 5^-} \dfrac{ 9 }{x - 5}?
    1. +∞+\infty
    2. −∞-\infty
    3. 00
  1. f(x)=x+8x2−9f(x) = \dfrac{x + 8}{x^2 - 9} has vertical asymptotes. What is the positive one, x=cx = c?
  2. f(x)=x+4x2−4f(x) = \dfrac{x + 4}{x^2 - 4} has vertical asymptotes. What is the positive one, x=cx = c?

Exit ticket

Three questions to check you have got it.

  1. What is lim⁡x→8−6x−8\displaystyle\lim_{x \to 8^-} \dfrac{ 6 }{x - 8}?
    1. +∞+\infty
    2. −∞-\infty
    3. 00
  2. What is lim⁡x→2−7x−2\displaystyle\lim_{x \to 2^-} \dfrac{ 7 }{x - 2}?
    1. +∞+\infty
    2. −∞-\infty
    3. 00
  3. What is lim⁡x→5−9x−5\displaystyle\lim_{x \to 5^-} \dfrac{ 9 }{x - 5}?
    1. +∞+\infty
    2. −∞-\infty
    3. 00

Summary card

Infinite limits

  • Denominator → 0, numerator not: ±∞\pm\infty, a vertical asymptote.
  • Test a value just beside aa for the sign.

Watch out: left and right of the asymptote, the signs can differ.

End of the lesson

Next lesson: Continuity Back to the chapter

Answers: Infinite limits and vertical asymptotes

A

  1. +∞+\infty
  2. +∞+\infty
  3. +∞+\infty

B

  1. −∞-\infty
  2. −∞-\infty
  3. −∞-\infty

C

  1. 33
  2. 22

D

  1. −∞-\infty
  2. −∞-\infty
  3. −∞-\infty

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