Find the limit, if it exists. lim x→−8 5x + 40 |x + 8|

Recall that



For the limit to exist, the limits from either side must also exist and match up. When , we're considering the domain , which corresponds to the second case in the piecewise definition above, i.e. . So we have



On the other hand, as , we're assuming , so that , in which case



The limits do not match because they differ by their sign, so the original limit does not exist.