Section 1.1 The Main Idea
Consider the function
\begin{equation*}
sin(x)/x
\end{equation*}
Notice that this function can’t be evaluated at x=0 (i.e. 0 is not in its domain.) But, we could compute it for values of x *close* to 0. For example, consider the following table for sin(x)/x.
| 0.500 |
0.958851 |
| 0.100 |
0.998334 |
| 0.010 |
0.999983 |
| 0.001 |
0.999999 |
Look at this table. As we move the \(x\) values closer to \(0\text{,}\) what are the \(sin(x)/x\) values getting closer to? It’s apparent that as \(x\) approaches \(0\text{,}\) that \(sin(x)/x\) approaches \(1\text{.}\) Of course, to be fair, at the moment we’re only letting \(x\) approach \(0\) from the right hand (positive) side of \(0\text{.}\) Let’s create some notation to record this kind of information.
Definition 1.1.1.
If \(f(x)\) approaches a number, say \(l\text{,}\) as \(x\) approaches a number \(a\) from its right hand side, then we write
\begin{gather*}
\displaystyle \lim_{x\to a^{+}}f(x)=l
\end{gather*}
which is pronounced "The limit of \(f(x)\) is \(l\) as \(x\) approaches \(a\) from the right". We also say that \(f(x)\) has a right hand limit of \(l\) at \(a\text{.}\) Analogously, If \(f(x)\) approaches a number, say \(k\) as \(x\) approaches \(a\) from its left hand side, then we write
\begin{gather*}
\displaystyle \lim_{x\to a^{-}}f(x)=k
\end{gather*}
which is pronounced "The limit of \(f(x)\) is \(k\) as \(x\) approaches \(a\) from the left". Again, we also say that \(f(x)\) has a left hand limit of \(k\) at \(a\text{.}\)
So now, with this new notation, it seems reasonable to write
\begin{gather*}
\displaystyle \lim_{x\to 0^{+}}(sin(x)/x)=1
\end{gather*}
Remark 1.1.2.
It is critical to note here that we are using the word "approach" in the definition above to indicate that we simply take values of the domain variable \(x\) *close* to \(a\) but not equal to it. In computing limits, we do not use \(x=a\text{.}\) We’re just identifying what \(f(x)\) is becoming as \(x\) *approaches* \(a\text{.}\) In fact, in many cases such as \(sin(x)/x\text{,}\) we can’t even compute the function at \(x=a=0\text{.}\)
Example 1.1.3.
For \(f(x)=sin(x)/x\) from above, make a table like the one above but this time use 4 or 5 domain values approaching \(0\) from the left. Use the notation from the definition above to write your conclusion about the left hand limit.Finally, since it will often be the case that both the left hand and right hand limit come out the same (but not always) for a function, we just drop the \(+\) and \(-\) in that case. In other words ...
Definition 1.1.4.
In case both the left hand and right hand limit of a function are the same (i.e. \(\displaystyle \lim_{x\to a^{+}}f(x)=\displaystyle \lim_{x\to a^{-}}f(x)=l\)), then we simply write \(\displaystyle \lim_{x\to a}f(x)=l\) without any reference to \(+\) or \(-\text{.}\) In this case we say that \(f(x)\) has a limit of \(l\) at \(a\text{.}\)
Example 1.1.5.
For \(f(x)=|x|/x\text{,}\) notice that once again \(0\) is not in its domain. Make tables for left and right hand approaches to \(0\) and use proper notation as defined above to write your conclusions about right hand limit, left hand limit and limit.
