Infinite limits - vertical asymptotes

Lessons

i) ${\;}\lim_{x \to {a^ - }} f\left( x \right) =\infty$	ii) $\lim_{x \to {a^ + }} f\left( x \right) =\infty$	iii) $\lim_{x \to {a^ - }} f\left( x \right) =,- \infty$	iv) $\lim_{x \to {a^ + }} f\left( x \right) =,- \infty$

• Introduction
  Introduction to Vertical Asymptotes
  
  a)

  finite limits VS. infinite limits
  
  
  b)

  infinite limits translate to vertical asymptotes on the graph of a function
  
  
  c)

  vertical asymptotes and curve sketching
  
  

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• 1.
  Determine Infinite Limits Graphically
  
  For the function $f$ whose graph is shown, state the following:
  a)

  $\lim_{x \to - {4^ - }} \;f\left( x \right)$
  $\lim_{x \to - {4^ + }} \;f\left( x \right)$
  $\lim_{x \to - 4} \;f\left( x \right)$
  
  
  b)

  $\lim_{x \to {1^ - }} \;f\left( x \right)$
  $\lim_{x \to {1^ + }} \;f\left( x \right)$
  $\lim_{x \to 1} \;f\left( x \right)$
  
  
  c)

  $\lim_{x \to {3^ - }} \;f\left( x \right)$
  $\lim_{x \to {3^ + }} \;f\left( x \right)$
  $\lim_{x \to 3} \;f\left( x \right)$
  
  
  d)

  $\lim_{x \to {5^ - }} \;f\left( x \right)$
  $\lim_{x \to {5^ + }} \;f\left( x \right)$
  $\lim_{x \to 5} \;f\left( x \right)$
  
  

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• 2.
  Evaluate Infinite Limits Algebraically
  Find:
  a)

  $\lim_{x \to {0^ - }} \;\frac{1}{x}$
  $\lim_{x \to {0^ + }} \;\frac{1}{x}$
  $\lim_{x \to 0} \;\frac{1}{x}$
  
  
  b)

  $\lim_{x \to {0^ - }} \;\frac{1}{{{x^2}}}$
  $\lim_{x \to {0^ + }} \;\frac{1}{{{x^2}}}$
  $\lim_{x \to 0} \;\frac{1}{{{x^2}}}$
  
  

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• 3.
  Evaluate Limits Algebraically
  Find:
  $\lim_{x \to {2^ - }} \;\frac{{5x}}{{x - 2}}$
  $\lim_{x \to {2^ + }} \;\frac{{5x}}{{x - 2}}$
  $\lim_{x \to 2} \;\frac{{5x}}{{x - 2}}$

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• 4.
  Determine Infinite Limits of Log Functions
  Determine:
  $\lim_{x \to {0^ + }} \ln x$

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