Factoring Polynomials

This works best with polynomials in a fractional form and in the indeterminate form $\frac{0}{0}$ . If the polynomial can be factored out so its multiples that are causing the denominator and numerator to take the form $\frac{0}{0}$ can be canceled out.

Given $f (x) = \frac{x^{2} + x - 6}{x^{2} - 4}$ if asked to find $lim_{x \to 2} f (x)$ you can factor out the polynomial as such:

\begin{aligned} lim_{x \to 2} \frac{x^{2} + x - 6}{x^{2} - 4} & = l i m_{x \to 2} \frac{(x - 2) (x + 3)}{(x - 2) (x + 2)} \\ = lim_{x \to 2} \frac{x + 3}{x + 2} \end{aligned}

Direct Substitution can then be used to solve the limit:

lim_{x \to 2} \frac{x + 3}{x + 2} = \frac{2 + 3}{2 + 2} = \frac{5}{4}