Question Video: Finding the One-Sided Limits of a Piecewise-Defined Function Mathematics

Video Transcript

Find the limit of 𝑓 of 𝑥 as 𝑥 approaches negative nine from the left and the limit of 𝑓 of 𝑥 as 𝑥 approaches negative nine from the right, where 𝑓 of 𝑥 is defined to be 78 if 𝑥 is less than negative nine and negative nine 𝑥 minus seven if 𝑥 is greater than or equal to negative nine.

𝑓 is a piecewise defined function. And we’re looking for the left-hand and right-hand limits of 𝑓 of 𝑥 as 𝑥 approaches the value negative nine for which the rule defining the function changes. Let’s start with the left-hand limit, the limit of 𝑓 of 𝑥 as 𝑥 approaches negative nine from the left.

It’s this superscript minus sign by the negative nine which tells us that we’re approaching from the left. That is, we’re only considering values of 𝑥 which are less than negative nine. And we can see that, for any value of 𝑥 less than negative nine, 𝑓 of 𝑥 is just 78.

Because for this limit we don’t care what 𝑓 of 𝑥 does when 𝑥 is greater than or equal to negative nine, we’re approaching from the left. And so 𝑥 is less than negative nine for our limit. We can just replace 𝑓 of 𝑥 by 78, which is how it is defined when 𝑥 is less than negative nine.

And the limit of a constant function like 78 is just that constant. So the limit is just 78. That’s the left-hand limit evaluated. What about the right-hand limit? Now we have this superscript plus sign, which tells us that 𝑥 is approaching negative nine from the right. So 𝑥 is greater than negative nine, but getting closer and closer to negative nine.

And so for this limit, we only care what the function is doing when 𝑥 is greater than negative nine. When 𝑥 is greater than negative nine, 𝑥 is certainly greater than or equal to negative nine. And 𝑓 of 𝑥 is just negative nine 𝑥 minus seven in this region.

This is now the limit of a polynomial function, in fact a linear function. And so we can just directly substitute to find the value of this limit. We substitute in negative nine and evaluate this expression to get 74.

Putting it all together then, we see that the value of the left-hand limit, the limit of 𝑓 of 𝑥 as 𝑥 approaches negative nine from the left, is 78. And the value of the right-hand limit, the limit of 𝑓 of 𝑥 as 𝑥 approaches negative nine from the right, is 74.

Just as an aside, we see that as the left-hand and right-hand limits have a different value, the left-hand limit is 78, whereas the right-hand limit is 74. The limit of 𝑓 of 𝑥 as 𝑥 approaches negative nine, period, does not exist.