Section 1.1 The Main Idea
\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.
\(f(x)\)\(l\text{,}\)\(x\)\(a\)
\begin{gather*}
\displaystyle \lim_{x\to a^{+}}f(x)=l
\end{gather*}
\(f(x)\)\(l\)\(x\)\(a\)\(f(x)\)\(l\)\(a\text{.}\)\(f(x)\)\(k\)\(x\)\(a\)left
\begin{gather*}
\displaystyle \lim_{x\to a^{-}}f(x)=k
\end{gather*}
\(f(x)\)\(k\)\(x\)\(a\)\(f(x)\)\(k\)\(a\text{.}\)
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.
\(x\)\(a\)\(x=a\text{.}\)\(f(x)\)\(x\)\(a\text{.}\)\(sin(x)/x\text{,}\)\(x=a=0\text{.}\)
Example 1.1.3.
\(f(x)=sin(x)/x\)\(0\)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.
\(\displaystyle \lim_{x\to a^{+}}f(x)=\displaystyle \lim_{x\to a^{-}}f(x)=l\)\(\displaystyle \lim_{x\to a}f(x)=l\)\(+\)\(-\text{.}\)\(f(x)\)\(l\)\(a\text{.}\)
Example 1.1.5.
\(f(x)=|x|/x\text{,}\)\(0\)\(0\)
