Infinite limits - vertical asymptotes

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Introduction
Lessons
  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
Examples
Lessons
  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
    Find:
    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
    Find:
    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
      Determine:
      limx0+lnx\lim_{x \to {0^ + }} \ln x
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      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.
      i)
        limxaf(x)={\;}\lim_{x \to {a^ - }} f\left( x \right) =\infty
      ii)
      limxa+f(x)=\lim_{x \to {a^ + }} f\left( x \right) =\infty
      iii)
      limxaf(x)=,\lim_{x \to {a^ - }} f\left( x \right) =,- \infty
      iv)
      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^+