Functions and Limits: Limits, One-Sided Limits, and Vertical Asymptotes

Definition of a Limit

  • Suppose a function ff is defined for all xx in an open interval containing aa, except possibly at x=ax = a.
  • If f(x)f(x) is arbitrarily close to a number LL for all xx sufficiently close to aa (with x≠ax \neq a), then the limit of f(x)f(x) as xx approaches aa equals LL: lim⁡x→af(x)=L\lim_{x \to a} f(x) = L
  • Example limit: lim⁡x→1x−1x2−1=0.5\lim_{x \to 1} \frac{x - 1}{x^2 - 1} = 0.5

One-Sided and Two-Sided Limits

  • Right-Sided Limit: Written as lim⁡x→a+f(x)=L\lim_{x \to a^+} f(x) = L if f(x)f(x) is arbitrarily close to LL for all xx near aa with x>ax > a.
  • Left-Sided Limit: Written as lim⁡x→a−f(x)=L\lim_{x \to a^-} f(x) = L if f(x)f(x) is arbitrarily close to LL for all xx near aa with x<ax < a.
  • Two-Sided Limit Condition: The limit lim⁡x→af(x)=L\lim_{x \to a} f(x) = L exists if and only if both one-sided limits exist and are equal: lim⁡x→a+f(x)=L\lim_{x \to a^+} f(x) = Llim⁡x→a−f(x)=L\lim_{x \to a^-} f(x) = L
  • Non-Existence of Limits: A limit lim⁡x→af(x)\lim_{x \to a} f(x) does not exist if a one-sided limit does not exist, or if: lim⁡x→a+f(x)≠lim⁡x→a−f(x)\lim_{x \to a^+} f(x) \neq \lim_{x \to a^-} f(x)
  • Example (Heaviside function): H(t)={0if t<01if t≥0H(t) = \begin{cases} 0 & \text{if } t < 0 \\ 1 & \text{if } t \ge 0 \end{cases}lim⁡t→0+H(t)=1\lim_{t \to 0^+} H(t) = 1lim⁡t→0−H(t)=0\lim_{t \to 0^-} H(t) = 0

Infinite Limits

  • Positive Infinite Limit: Written as lim⁡x→af(x)=∞\lim_{x \to a} f(x) = \infty or f(x)→∞f(x) \to \infty as x→ax \to a when f(x)f(x) becomes arbitrarily large as xx approaches aa (with x≠ax \neq a).
  • Negative Infinite Limit: Written as lim⁡x→af(x)=−∞\lim_{x \to a} f(x) = -\infty or f(x)→−∞f(x) \to -\infty as x→ax \to a when f(x)f(x) becomes arbitrarily large negative as xx approaches aa (with x≠ax \neq a).
  • Non-Existence Example: The limit lim⁡x→01x2\lim_{x \to 0} \frac{1}{x^2} does not exist because values grow without bound.
  • Negative Infinite Limit Example: lim⁡x→0(−1x2)=−∞\lim_{x \to 0} \left(-\frac{1}{x^2}\right) = -\infty
  • One-Sided Infinite Limits: lim⁡x→a−f(x)=∞\lim_{x \to a^-} f(x) = \inftylim⁡x→a+f(x)=∞\lim_{x \to a^+} f(x) = \inftylim⁡x→a−f(x)=−∞\lim_{x \to a^-} f(x) = -\inftylim⁡x→a+f(x)=−∞\lim_{x \to a^+} f(x) = -\infty

Vertical Asymptotes

  • The line x=ax = a is a vertical asymptote of the curve y=f(x)y = f(x) if at least one of the following statements holds: lim⁡x→af(x)=∞\lim_{x \to a} f(x) = \inftylim⁡x→a−f(x)=∞\lim_{x \to a^-} f(x) = \inftylim⁡x→a+f(x)=∞\lim_{x \to a^+} f(x) = \inftylim⁡x→af(x)=−∞\lim_{x \to a} f(x) = -\inftylim⁡x→a−f(x)=−∞\lim_{x \to a^-} f(x) = -\inftylim⁡x→a+f(x)=−∞\lim_{x \to a^+} f(x) = -\infty
  • Example: The curve y=2xx−3y = \frac{2x}{x - 3} has a vertical asymptote at: x=3x = 3