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\begin{frame} \frametitle{Continuity} \begin{block}{} These functions are continuous at each point of their domain: \begin{center} polynomials \hspace{1cm} rationals \hspace{1cm} root functions\\[1ex] (inverse) trigonometric \hspace{1cm} exponential \hspace{1cm} logarithmic \end{center} \end{block} \pause\medskip \begin{block}{} Inverse functions of continuous functions are continuous. \end{block} \pause\medskip Recall that continuity at $a$ means that \begin{align*} \lim_{x\to a} f(x) = f(a) \end{align*} and this is \emph{direct substitution}. \pause\medskip \begin{exampleblock}{} Evaluate $\lim_{x\to \pi} f(x)$ where $f(x) = \frac{\sin x}{2 + \cos x}$. \pause\\[1ex] We know that $\sin$, $\cos$ and $2$ are continuous functions.\\\pause Then their sum and quotient are continuous on their domain.\\\pause The domain contains $\pi$, so: $\lim_{x\to \pi} f(x) = f(\pi) = 0/(2-1) = 0$. \end{exampleblock} \end{frame}