Video Transcript
Determine the limit as π₯ tends to four of π of π₯, if it exists.
We have a graph of π of π₯ and we want to use it to see what happens as π₯ tends to four. We can see from this point here that π of four is equal to one. But this doesnβt help us because weβre looking for the limit as π₯ tends to four. And so, we look at the value of the function π when π₯ is near but not equal to four. When π₯ is two, π of π₯ is one. When π₯ is closer to four, π₯ is three, π of π₯ is zero. And we can get closer still, when π₯ is 3.5, π of π₯ is negative 0.5. We can see that as π₯ gets closer and closer to four from below, π of π₯ gets closer and closer to negative one.
And we can write down this fact like this: the limit as π₯ tends to four from below of π of π₯ is equal to negative one. So weβve looked at what happens when π₯ approaches four from below, that is π₯ is less than four but getting closer and closer to four.
Now we need to look at what happens when π₯ approaches four from above. π of six is one. As π₯ gets closer to four, π of π₯ gets closer to negative one. So π of five is closer to negative one than π of six was; π of five is zero. And this trend continues as π₯ approaches four from above, that is π₯ is greater than four but getting closer and closer to four. We can see that π of π₯ gets closer and closer to negative one. And so the limit as π₯ tends to four from above of π of π₯ is also equal to negative one.
The limits from above and from below are equal. And so we can say that the limit as π₯ tends to four, period, of π of π₯ exists and is equal to negative one. Notice however that the limit as π₯ tends to four of π of π₯, which we found was negative one, is not equal to the value of π of four which, according to the graph, is one. So the limit of a function at a point can exist but not be equal to the value of the function at that point.
