Video Transcript
Using the graph shown, determine
the limit as π₯ tends to three of π of π₯.
Here, we have the graph of π of
π₯. And weβve been asked to find the
limit as π₯ tends to three. We can see that at π₯ is equal to
three, π of π₯ is defined to be negative five. However, when we are finding the
limit of a function at a particular point, the value of that function at that point
does not matter. What matters is whatβs happening to
the function around that point. This is because the limit as π₯
approaches three of π of π₯ is defined to be the value π of π₯ approaches as π₯
tends to three. Letβs consider whatβs happening to
π of π₯ to the left and to the right of π₯ is equal to three.
If we consider π of π₯ to the left
of π₯ is equal to three, we can see that π of π₯ is increasing and getting closer
and closer to the value of two, as π₯ is getting closer and closer to three. And as π₯ approaches three from the
right, π of π₯ is again increasing. And it is also getting closer and
closer to the value of two. Now, this tells us what we need to
know about π of π₯ as π₯ tends to three, since from both the left and right, the
value of π of π₯ approaches two as π₯ approaches three. And so, even though the value of π
of three is equal to negative five, the limit as π₯ approaches three of π of π₯ is
equal to negative two, which is our solution to this question. In this previous example, weβve
seen how even though a function may be defined at a different point at a particular
π₯-value, the limit as π₯ approaches that particular π₯-value of π of π₯ may be
different to the value of π of π₯ at that point.
