What is the correct notation that
describes the following statement? As 𝑥 approaches zero, 𝑓 of 𝑥
approaches negative six.
For a question of this type, the
first thing we might notice is this word approaches. Whenever we are told that the value
of a function or a variable approaches something, it gives us a hint that our
question might involve limits. The standard notation for a limit
is shown here. And we would read the statement as:
the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 is equal to 𝐿. Breaking it down, what this
statement is telling us is that as the value of 𝑥 approaches the constant, here
called 𝑎, the value of 𝑓 of 𝑥 will approach 𝐿. This 𝐿 is sometimes referred to as
the value of the limit. A quick tip here is to remember
that we’re not being told that 𝑓 of 𝑥 is equal to 𝐿, but rather that the value of
our limit is equal to 𝐿.
Okay, now that we understand how to
write a limit and how to interpret the statement, let’s see how it applies to our
question. We are told that as 𝑥 approaches
zero, 𝑓 of 𝑥 approaches negative six. This zero is the value which we
consider 𝑥 to be approaching. And it’s represented by 𝑎 in our
general limit equation. Similarly, this negative six, which
is the value 𝑓 of 𝑥 approaches, is represented by 𝐿 in our general limit
equation. We therefore write the following
statement. The limit as 𝑥 approaches zero of
𝑓 of 𝑥 is equal to negative six. Since we have substituted 𝑎 is
zero and 𝐿 is negative six, the statement is interpreted in the following way. As the value of 𝑥 approaches zero,
the value of 𝑓 of 𝑥 approaches negative six.
Looking at the statement in the
question and the statement that we have just written, these exactly match. This means that we have expressed
the statement given in the question using the correct notation for a limit.