A piece-wise function is made up of several sub-functions, where each sub-function corresponds to a piece of the domain. In this example, everywhere where [math]x\ne4[/math], [math]f\left(x\right)=x^2-6[/math]. However, where [math]x=4[/math], [math]f\left(x\right)=1[/math]. [br][br]To find the limit at [math]x[/math] approaches 4, we don't need to focus on the value of [math]f\left(4\right)[/math], but instead the values of the function [math]f\left(x\right)=x^2-6[/math] near 4.

So let's focus only on the part of the function [i]around [math]x=4[/math]. [/i]This means that we will deal only with [math]f\left(x\right)=x^2-6[/math].

We can see from the graph above that as [math]x[/math] nears 4, [math]f\left(x\right)[/math] nears 10. [br][br][br]Solution: Therefore, [math]\lim_{x\rightarrow4}[/math][math]x^2-6=10.[/math]

Symbolab.com is like wolframalpha, but free. Don't use it unless you're stuck so you don't become dependent on it![br][br]Desmos.com: graphing calculator[br]Betterexplained.com[br]Mathinsight.org