 Every now and then we run into a problem where we have to deal with what are called one-sided limits. In general, when we try to find the limit as x approaches a of f of x, we try to evaluate f of x for x values that are close to a. So we might look at x values that are a little bit more than a, or we might look at x values that are a little less than a. However, it may be convenient or necessary to focus on just one of these cases. And this leads to what are called one-sided limits. So a little bit of notation. We write limit as x approaches a plus f of x to indicate the limit of f of x as x approaches a, but always stays a little bit greater than a. And we might say from above. Similarly, we write limit as x approaches a minus f of x to indicate the limit of f of x as x approaches a, but always staying a little less than a. And we might say from below. If the limit from above and the limit from below are the same, then the limit without a plus or minus is the common value. Otherwise, the limit doesn't exist. And it's important to emphasize that the unqualified limit exists if and only if the one-sided limits exist, and they agree. This type of problem often shows up when we're dealing with functions that are defined piecewise. So let's take a look at the limit as x approaches four from below, the limit as x approaches four from above, and the limit as x approaches four unqualified. So as long as x is less than four, our function is going to be x squared. So the limit as x approaches four from below of f of x is going to be the same as the limit as x approaches four from below of x squared, which will be 16. Similarly, if x is greater than four, then our function is 12 minus 3x. So the limit as x approaches four from above of f of x is the limit as x approaches four from above of 12 minus 3x, which is zero. And finally, since the limit as x approaches four from below and the limit as x approaches four from above are different, then our unqualified limit does not exist. Thank you.