Factor where possible, and simplify.

[latex]\begin{align}\underset{x\to 2}{\mathrm{lim}}\left(\frac{{x}^{2}-6x+8}{x - 2}\right)&=\underset{x\to 2}{\mathrm{lim}}\left(\frac{\left(x - 2\right)\left(x - 4\right)}{x - 2}\right)&& \text{Factor the numerator}. \\ &=\underset{x\to 2}{\mathrm{lim}}\left(\frac{\cancel{\left(x - 2\right)}\left(x - 4\right)}{\cancel{x - 2}}\right)&& \text{Cancel the common factors}. \\ &=\underset{x\to 2}{\mathrm{lim}}\left(x - 4\right)&& \text{Evaluate}. \\ &=2 - 4=-2 \end{align}[/latex]

Analysis of the Solution

When the limit of a rational function cannot be evaluated directly, factored forms of the numerator and denominator may simplify to a result that can be evaluated.

Notice, the function

[latex]f\left(x\right)=\frac{{x}^{2}-6x+8}{x - 2}[/latex]

is equivalent to the function

[latex]f\left(x\right)=x - 4,x\ne 2[/latex].

Notice that the limit exists even though the function is not defined at [latex]x=2[/latex].

Find the LCD for the denominators of the two terms in the numerator, and convert both fractions to have the LCD as their denominator.

Analysis of the Solution

When determining the limit of a rational function that has terms added or subtracted in either the numerator or denominator, the first step is to find the common denominator of the added or subtracted terms; then, convert both terms to have that denominator, or simplify the rational function by multiplying numerator and denominator by the least common denominator. Then check to see if the resulting numerator and denominator have any common factors.