Finding limits algebraically - when direct substitution is not possible

Lessons

1. Simplify Out "Zero Denominator" by Cancelling Common Factors

   Find $\lim_{x \to 3} \;\frac{{{x^2} - 9}}{{x - 3}}$

2. Expand First, Then Simplify Out "Zero Denominator" by Cancelling Common Factors

   Evaluate $\lim_{h \to 0} \;\frac{{{{\left( {5 + h} \right)}^2} - 25}}{h}$

3. Simplify Out "Zero Denominator" by Rationalizing Radicals

   Evaluate:

   1. $\lim_{x \to 4} \;\frac{{4 - x}}{{2 - \sqrt x }}$
      (hint: rationalize the denominator by multiplying its conjugate)
4. Find Limits of Functions involving Absolute Value

   Evaluate $\lim_{x \to 0} \;\frac{{\left| x \right|}}{x}$

   (hint: express the absolute value function as a piece-wise function)

5. Find Limits Using the Trigonometric Identity:$\lim_{\theta \to 0} \;\frac{{{sin\;}\theta}}{{\theta}}=1$

   Find $\lim_{x \to 0} \;\frac{{{sin\;}5x}}{{2x}}$