Video Transcript
Using the graph shown, determine
the limit as 𝑥 tends to three of 𝑓 of 𝑥.
Here, we have the graph of 𝑓 of
𝑥. And we’ve been asked to find the
limit as 𝑥 tends to three. We can see that at 𝑥 is equal to
three, 𝑓 of 𝑥 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 what’s happening to
the function around that point. This is because the limit as 𝑥
approaches three of 𝑓 of 𝑥 is defined to be the value 𝑓 of 𝑥 approaches as 𝑥
tends to three. Let’s consider what’s happening to
𝑓 of 𝑥 to the left and to the right of 𝑥 is equal to three.
If we consider 𝑓 of 𝑥 to the left
of 𝑥 is equal to three, we can see that 𝑓 of 𝑥 is increasing and getting closer
and closer to the value of two, as 𝑥 is getting closer and closer to three. And as 𝑥 approaches three from the
right, 𝑓 of 𝑥 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 𝑓 of 𝑥 as 𝑥 tends to three, since from both the left and right, the
value of 𝑓 of 𝑥 approaches two as 𝑥 approaches three. And so, even though the value of 𝑓
of three is equal to negative five, the limit as 𝑥 approaches three of 𝑓 of 𝑥 is
equal to negative two, which is our solution to this question. In this previous example, we’ve
seen how even though a function may be defined at a different point at a particular
𝑥-value, the limit as 𝑥 approaches that particular 𝑥-value of 𝑓 of 𝑥 may be
different to the value of 𝑓 of 𝑥 at that point.
