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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.

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.

Find the vertical asymptote(s) of f(x) = 1/(x² - 1).
  1. x² - 1 = (x-1)(x+1) = 0 when x = ±1
  2. Both x = 1 and x = -1 are vertical asymptotes

Answer: Vertical asymptotes at x = -1 and x = 1.

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