Video Transcript
The following figure represents the
graph of the function 𝑓 of 𝑥 is equal to 𝑥 squared. What does the graph suggest about
the value of the limit as 𝑥 approaches two of 𝑓 of 𝑥.
For this question, we have been
given a function. And we’ve been asked to evaluate a
limit of said function. To interpret this, let us recall
the general form of a limit. The limit as 𝑥 approaches 𝑎 of 𝑓
of 𝑥 is equal to 𝐿. What this statement tells us is
that the value of 𝑓 of 𝑥 will approach 𝐿 as the value of 𝑥 approaches 𝑎 from
both sides. But remember, we’re not concerned
with the point where 𝑥 is actually equal to 𝑎. Let us now apply this statement to
our question.
The limit as 𝑥 approaches two of
𝑓 of 𝑥 is equal to some value. And we’ll call this value 𝐿
one. This 𝐿 one is the thing we need to
find. What the statement is telling us is
that the value of 𝑓 of 𝑥 approaches 𝐿 one as the value of 𝑥 approaches two from
both sides. And remember, this doesn’t
necessarily have to be the same as the value of our function when 𝑥 is equal to
two. To find the value of 𝐿 one, we can
see what happens to the value of our function as we get closer and closer to 𝑥
equals two. We’ll start by considering some
value of 𝑥 which is slightly smaller than two.
Let’s say 𝑥 equals 1.5. In this case, 𝑓 of 𝑥 equals
2.25. Since we know that 𝑓 of 𝑥 equals
𝑥 squared, we could even verify our graphical findings by taking 1.5 squared. Let us now increase our value of
𝑥. So we’re approaching 𝑥 equals two
from the left, so when 𝑥 is less than two. As we do this, we might begin to
notice that the value of 𝑓 of 𝑥 appears to be approaching four. If we were to follow a similar
process, starting with the value of 𝑥 which is slightly larger than two and then we
were to approach 𝑥 equals two from the right. We would notice that the values of
𝑓 of 𝑥 also approach four.
Without really going into any
calculations, we see that as 𝑥 gets closer and closer to two, 𝑓 of 𝑥 gets closer
and closer to four. This is true if we approach from
the left or from the right, so from both sides. Essentially, on our graph, we’re
converging on the point two, four. Okay, so we have just shown that
the value of 𝑓 of 𝑥 approaches four as the value of 𝑥 approaches two. Since this statement defines our
limit, we can use our finding to say that the value of the limit is also four. In doing this, we have answered our
question. Using our graph, we observed that
the value of 𝑓 of 𝑥 approached four as the value of 𝑥 approached two from both
the left and the right. We use this to say that the limit
as 𝑥 approaches two of 𝑓 of 𝑥 is equal to four.
