Infinite limits - vertical asymptotes

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Intros
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
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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. lim⁡x→−4−  f(x)\lim_{x \to - {4^ - }} \;f\left( x \right)
      lim⁡x→−4+  f(x)\lim_{x \to - {4^ + }} \;f\left( x \right)
      lim⁡x→−4  f(x)\lim_{x \to - 4} \;f\left( x \right)
    2. lim⁡x→1−  f(x)\lim_{x \to {1^ - }} \;f\left( x \right)
      lim⁡x→1+  f(x)\lim_{x \to {1^ + }} \;f\left( x \right)
      lim⁡x→1  f(x)\lim_{x \to 1} \;f\left( x \right)
    3. lim⁡x→3−  f(x)\lim_{x \to {3^ - }} \;f\left( x \right)
      lim⁡x→3+  f(x)\lim_{x \to {3^ + }} \;f\left( x \right)
      lim⁡x→3  f(x)\lim_{x \to 3} \;f\left( x \right)
    4. lim⁡x→5−  f(x)\lim_{x \to {5^ - }} \;f\left( x \right)
      lim⁡x→5+  f(x)\lim_{x \to {5^ + }} \;f\left( x \right)
      lim⁡x→5  f(x)\lim_{x \to 5} \;f\left( x \right)
  2. Evaluate Infinite Limits Algebraically
    Find:
    1. lim⁡x→0−  1x\lim_{x \to {0^ - }} \;\frac{1}{x}
      lim⁡x→0+  1x\lim_{x \to {0^ + }} \;\frac{1}{x}
      lim⁡x→0  1x\lim_{x \to 0} \;\frac{1}{x}
    2. lim⁡x→0−  1x2\lim_{x \to {0^ - }} \;\frac{1}{{{x^2}}}
      lim⁡x→0+  1x2\lim_{x \to {0^ + }} \;\frac{1}{{{x^2}}}
      lim⁡x→0  1x2\lim_{x \to 0} \;\frac{1}{{{x^2}}}
  3. Evaluate Limits Algebraically
    Find:
    lim⁡x→2−  5xx−2\lim_{x \to {2^ - }} \;\frac{{5x}}{{x - 2}}
    lim⁡x→2+  5xx−2\lim_{x \to {2^ + }} \;\frac{{5x}}{{x - 2}}
    lim⁡x→2  5xx−2\lim_{x \to 2} \;\frac{{5x}}{{x - 2}}
    1. Determine Infinite Limits of Log Functions
      Determine:
      lim⁡x→0+ln⁡x\lim_{x \to {0^ + }} \ln x
      Topic Notes
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      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)
        lim⁡x→a−f(x)=∞{\;}\lim_{x \to {a^ - }} f\left( x \right) =\infty       		ii)
      lim⁡x→a+f(x)=∞\lim_{x \to {a^ + }} f\left( x \right) =\infty       		iii)
      lim⁡x→a−f(x)=,−∞\lim_{x \to {a^ - }} f\left( x \right) =,- \infty       		iv)
      lim⁡x→a+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^+