Limit Fundamentals

Cases

• \(\displaystyle \frac{\#}{0} = \left\{ \begin{array}{l} \infty\\-\infty\\ \texttt{DNE}\end{array}\right.\)
• \(\displaystyle \frac{0}{\#}=0\)
• \(\displaystyle \frac{0}{\pm \infty}=0\)

Intedeterminate Forms

• \(\displaystyle \frac{0}{0}\)
• \(\displaystyle \frac{\infty}{\infty}\)

1. Algebraic manipulate the limit
   1. Factor the numerator and the denominator of a rational function to divide out a common factor.
   2. Combine fractions into a single fraction and then simplify the results.
   3. When square roots are present you may wish to consider using the conjugate and applying the difference of squares identity.
   4. Consider how the Squeeze Theorem can be used to assess the limit.
2. Apply L’Hôpital’s Rule
   1. Only applicable if the limit corresponds to either \(\textstyle\frac{\infty}{\infty}\) or \(\textstyle\frac{0}{0}\)
   2. \(\underset{x\to a}{\lim}\frac{f(x)}{g(x)}\stackrel{\small{L’H}}{=}\underset{x\to a}{\lim}\frac{f'(x)}{g'(x)}\)