### Video Transcript

Which of the following statements
is not the same as saying that the limit as π₯ approaches eight of π of π₯ is equal
to three? Is it option (A) we can make π of
π₯ as close as we like to three by taking π₯ sufficiently close to eight? Is it option (B) π evaluated at
three is equal to π evaluated at eight? Option (C) as π₯ gets closer and
closer to eight, π of π₯ gets closer and closer to three. Or is it option (D) π of π₯
approaches three as π₯ approaches eight?

In this question, weβre given four
statements, and we need to determine which of these four statements is not the same
as saying that the limit as π₯ approaches eight of π of π₯ is equal to three. To answer this question, letβs
start by recalling what we mean by the value of the limit of a function at a
point. We recall that we say if the values
of π of π₯ approach some finite value of πΏ, as the values of π₯ approach some
value of π from either side but not necessarily when π₯ is equal to π, then we say
that the limit as π₯ approaches π of π of π₯ is equal to πΏ. And we can directly use this
definition to answer the question. However, letβs start by replacing
the values of π and πΏ in the definition with the values given in the question.

Weβre taking the limit as π₯
approaches eight of our function π of π₯, so weβll set our value of π equal to
eight. And we say that this limit is equal
to three, so weβll set our value of πΏ equal to three. We can then update our
definition. This now tells us since the limit
as π₯ approaches eight of π of π₯ is equal to three as the values of π₯ approach
eight from either side but not necessarily when π₯ is equal to eight, then the
values of π of π₯ must be approaching three. And this is really useful because
we can see that this statement is equivalent to three of our options.

First, we can make π of π₯ as
close as we like to three by taking π₯ sufficiently close to eight. This is a direct result from our
definition. Our values of π of π₯ are
approaching three as our values of π₯ approach eight. And since π of π₯ approaches
three, we can make this as close as we like by taking π₯ sufficiently close to
eight. So, option (A) is exactly the same
as saying the limit as π₯ approaches eight of π of π₯ is equal to three.

We can see the same is true in
option (C). This is exactly the same as saying
the limit as π₯ approaches eight of π of π₯ is equal to three. Itβs almost exactly the same as
what is written. We only need to note one thing. When we say that our values of π₯
are getting closer and closer to eight, we do mean that π₯ can approach from either
side and we donβt need to know what happens when π₯ is equal to eight. And this is exactly the same as our
definition. As our values of π₯ approach eight
from either side but not necessarily when π₯ is equal to eight, the values of π of
π₯ are approaching three.

Finally, in option (D), we see that
π of π₯ approaches three as π₯ approaches eight. And once again, this is exactly the
same as what is written. So, this option is also the same as
saying the limit as π₯ approaches eight of π of π₯ is equal to three. Now, this is enough to answer our
question. The other three options are all the
same. So, option (B) must be
different. However, for due diligence, letβs
show this.

This statement says that the value
of π of three needs to be equal to π evaluated at eight. Of course, we can immediately
notice a few things wrong. For example, this statement does
not tell us any information about the values of π of π₯ as our values of π₯ get
closer and closer to eight. And we can also notice in our
definition we donβt need to know the value of our function when π₯ is equal to
eight. The exact value of π evaluated at
eight will have no effect on the limit as π₯ approaches eight of π of π₯. In particular, our function does
not even need to be defined at π₯ is equal to eight.

So, knowing that the limit as π₯
approaches eight of π of π₯ is equal to three wonβt tell us any information about
the function evaluated at eight or the function evaluated at three. So, we can say that option (B)
saying that π evaluated at three is equal to π evaluated at eight is not the same
as saying that the limit as π₯ approaches eight of π of π₯ is equal to three.