Video Transcript
True or false: If the limit as 𝑥
approaches five of 𝑓 of 𝑥 is equal to negative three, then 𝑓 of five must be
equal to negative three.
For this question, we’re given
information in the form of a limit. So let us interpret this
statement. What this is telling us is that as
the value of 𝑥 approaches five, the value of 𝑓 of 𝑥 approaches negative
three. Let us now look at the second part
of our question. The question is asking us if the
limit guarantees that when 𝑥 equals five, the value of the function will be
negative three. To answer this, let us recall the
general form of a limit.
We recall that the limit concerns
values of 𝑥 which are arbitrarily close to 𝑎 but not when 𝑥 is equal to 𝑎. In our question, this 𝑎 is
represented by five. The limit gives us information
about our function as 𝑥 approaches five but does not give us information about our
function when 𝑥 is equal to five. In fact, 𝑓 of five could be equal
to our limit, which is negative three. It could take any other value or it
could even be undefined. Since we cannot guarantee that the
value of 𝑓 of five is equal to negative three based on the limit, the answer to the
question is false. 𝑓 of five does not have to be
equal to negative three.
