How to find limits near vertical asymptotes

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Introduction

Lessons

Introduction to Vertical Asymptotes
finite limits VS. infinite limits
infinite limits translate to vertical asymptotes on the graph of a function
vertical asymptotes and curve sketching

Examples

Lessons

Determine Infinite Limits Graphically

For the function $f$ $f$ whose graph is shown, state the following:
1. $\lim_{x \to - {4^ - }} \;f\left( x \right)$
  $\lim_{x \to - {4^ + }} \;f\left( x \right)$
  $\lim_{x \to - 4} \;f\left( x \right)$
2. $\lim_{x \to {1^ - }} \;f\left( x \right)$
  $\lim_{x \to {1^ + }} \;f\left( x \right)$
  $\lim_{x \to 1} \;f\left( x \right)$
3. $\lim_{x \to {3^ - }} \;f\left( x \right)$
  $\lim_{x \to {3^ + }} \;f\left( x \right)$
  $\lim_{x \to 3} \;f\left( x \right)$
4. $\lim_{x \to {5^ - }} \;f\left( x \right)$
  $\lim_{x \to {5^ + }} \;f\left( x \right)$
  $\lim_{x \to 5} \;f\left( x \right)$
Evaluate Infinite Limits Algebraically
Find:
1. $\lim_{x \to {0^ - }} \;\frac{1}{x}$
  $\lim_{x \to {0^ + }} \;\frac{1}{x}$
  $\lim_{x \to 0} \;\frac{1}{x}$
2. $\lim_{x \to {0^ - }} \;\frac{1}{{{x^2}}}$
  $\lim_{x \to {0^ + }} \;\frac{1}{{{x^2}}}$
  $\lim_{x \to 0} \;\frac{1}{{{x^2}}}$
Evaluate Limits Algebraically
Find:
$\lim_{x \to {2^ - }} \;\frac{{5x}}{{x - 2}}$ $lim_{x \to 2^{-}} \frac{5 x}{x - 2}$
$\lim_{x \to {2^ + }} \;\frac{{5x}}{{x - 2}}$ $lim_{x \to 2^{+}} \frac{5 x}{x - 2}$
$\lim_{x \to 2} \;\frac{{5x}}{{x - 2}}$ $lim_{x \to 2} \frac{5 x}{x - 2}$
Determine Infinite Limits of Log Functions
Determine:
$\lim_{x \to {0^ + }} \ln x$ $lim_{x \to 0^{+}} ln x$

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

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