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Limits & Continuity · Continuity & Theorems

Infinite Limits (Vertical Asymptotes)

lim⁡x→af(x)=±∞⇒x=a is a vertical asymptote\lim_{x \to a} f(x) = \pm\infty \Rightarrow x = a \text{ is a vertical asymptote}

If f(x) approaches ±∞ as x approaches a, then the line x = a is a vertical asymptote of f.

  • 2 worked examples
  • 18 in Limits & Continuity

Worked examples

Find the vertical asymptote(s) of f(x) = 1/(x-3).
  1. Denominator = 0 when x = 3
  2. lim(x→3⁺) 1/(x-3) = +∞ and lim(x→3⁻) 1/(x-3) = -∞

Answer: Vertical asymptote at x = 3.

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Find the vertical asymptote(s) of f(x) = 1/(x² - 1).

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Common questions

What is the Infinite Limits (Vertical Asymptotes)?

If f(x) approaches ±∞ as x approaches a, then the line x = a is a vertical asymptote of f. It is one of the limits & continuity formulas in the CalcRef reference.

How do you use the Infinite Limits (Vertical Asymptotes)?

Worked example. Find the vertical asymptote(s) of f(x) = 1/(x-3). Denominator = 0 when x = 3. lim(x→3⁺) 1/(x-3) = +∞ and lim(x→3⁻) 1/(x-3) = -∞. Answer: Vertical asymptote at x = 3.

Practice

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