Infinite limits - vertical asymptotes

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  1. Introduction to Vertical Asymptotes
  2. finite limits VS. infinite limits
  3. infinite limits translate to vertical asymptotes on the graph of a function
  4. vertical asymptotes and curve sketching
  1. Determine Infinite Limits Graphically
    Finding limits algebraically using direct substitution
    For the function ff whose graph is shown, state the following:
    1. limx4  f(x)\lim_{x \to - {4^ - }} \;f\left( x \right)
      limx4+  f(x)\lim_{x \to - {4^ + }} \;f\left( x \right)
      limx4  f(x)\lim_{x \to - 4} \;f\left( x \right)
    2. limx1  f(x)\lim_{x \to {1^ - }} \;f\left( x \right)
      limx1+  f(x)\lim_{x \to {1^ + }} \;f\left( x \right)
      limx1  f(x)\lim_{x \to 1} \;f\left( x \right)
    3. limx3  f(x)\lim_{x \to {3^ - }} \;f\left( x \right)
      limx3+  f(x)\lim_{x \to {3^ + }} \;f\left( x \right)
      limx3  f(x)\lim_{x \to 3} \;f\left( x \right)
    4. limx5  f(x)\lim_{x \to {5^ - }} \;f\left( x \right)
      limx5+  f(x)\lim_{x \to {5^ + }} \;f\left( x \right)
      limx5  f(x)\lim_{x \to 5} \;f\left( x \right)
  2. Evaluate Infinite Limits Algebraically
    1. limx0  1x\lim_{x \to {0^ - }} \;\frac{1}{x}
      limx0+  1x\lim_{x \to {0^ + }} \;\frac{1}{x}
      limx0  1x\lim_{x \to 0} \;\frac{1}{x}
    2. limx0  1x2\lim_{x \to {0^ - }} \;\frac{1}{{{x^2}}}
      limx0+  1x2\lim_{x \to {0^ + }} \;\frac{1}{{{x^2}}}
      limx0  1x2\lim_{x \to 0} \;\frac{1}{{{x^2}}}
  3. Evaluate Limits Algebraically
    limx2  5xx2\lim_{x \to {2^ - }} \;\frac{{5x}}{{x - 2}}
    limx2+  5xx2\lim_{x \to {2^ + }} \;\frac{{5x}}{{x - 2}}
    limx2  5xx2\lim_{x \to 2} \;\frac{{5x}}{{x - 2}}
    1. Determine Infinite Limits of Log Functions
      limx0+lnx\lim_{x \to {0^ + }} \ln x
      Topic Notes
      Limits don't always necessarily give numerical solutions. What happens if we take the limit of a function near its vertical asymptotes? We will answer this question in this section, as well as exploring the idea of infinite limits using one-sided limits and two-sided limits.
        limxaf(x)={\;}\lim_{x \to {a^ - }} f\left( x \right) =\infty
      limxa+f(x)=\lim_{x \to {a^ + }} f\left( x \right) =\infty
      limxaf(x)=,\lim_{x \to {a^ - }} f\left( x \right) =,- \infty
      limxa+f(x)=,\lim_{x \to {a^ + }} f\left( x \right) =,- \infty
      Infinite limits - vertical asymptotes, x approaching a^- Infinite limits - vertical asymptotes, x approaching a^+ Infinite limits - vertical asymptotes, x approaching a^- Infinite limits - vertical asymptotes, x approaching a^+