Theorem 2.2.1.
Rational functions, \(e^x\text{,}\) \(ln(x)\text{,}\) \(sin(x)\) and \(cos(x)\) are all continuous wherever they are defined.
Section 2.2 Basic Properties
In light of the definition of continuity from the previous section, we see that Corollary 1.3.7 and Theorem 1.3.8 are really saying ...
The basic limit properties then yield the following
Corollary 2.2.2.
If \(f(x)\) and \(g(x)\) are both continuous at a given \(x\text{,}\) then \(rf(x)+sg(x)\) and \(f(x)g(x)\) are all continuous at that \(x\text{.}\) Further, if also \(g(x)\ne 0\) then \(f(x)/g(x)\) is continuous at \(x\text{.}\)
So, for example, if we want to calculate the limit of \(tan(x)\) at \(7.2\text{,}\) it’s very easy since we know from the corollary above that \(tan(x)\text{,}\) which is simply \(sin(x)/cos(x)\) is continuous at \(7.2\text{.}\) Thus
\begin{equation*}
\displaystyle \lim_{x\to 7.2}tan(x)=tan(7.2)
\end{equation*}
