Video Transcript
True or False: The limit as 𝑥
approaches 𝑎 of 𝑓 of 𝑥 is equal to 𝑙 means that as 𝑥 gets closer to 𝑎, 𝑓 of
𝑥 gets closer to 𝑙.
In this question, we need to
determine whether a statement is true or false. We need to determine whether saying
that the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 is equal to 𝑙 means that as 𝑥 gets
closer to 𝑎, 𝑓 of 𝑥 gets closer to 𝑙. And to answer this question, let’s
start by recalling what we mean by this limit notation. We can recall that we say if the
values of 𝑓 of 𝑥 approach some finite value of 𝑙 as the values of 𝑥 approach 𝑎
from either side, but not necessarily when 𝑥 is equal to 𝑎, then we say that the
limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 is equal to 𝑙. This is the definition of what this
limit notation means. And we can rewrite and reorder this
information slightly.
For example, saying that 𝑥
approaches 𝑎 from either side but not necessarily when 𝑥 is equal to 𝑎 is exactly
the same as saying our values of 𝑥 are getting closer and closer to 𝑎. Similarly, saying that when this
happens the values of 𝑓 of 𝑥 are approaching some finite value of 𝑙 is exactly
the same as saying our values of 𝑓 of 𝑥 are getting closer and closer to this
value of 𝑙. So this statement is exactly the
same as our definition; we’ve just reordered the information slightly.
Therefore, we can say it’s true
that saying the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 is equal to 𝑙 means that as
𝑥 gets closer to 𝑎, 𝑓 of 𝑥 gets closer to 𝑙.
