\begin{frame} \frametitle{Finding a Tangent} Alternative definition of the slope: \begin{block}{} The slope of $f$ at point $(a,f(a))$ is: \begin{talign} m = \lim_{h\to 0} \frac{f(a+h) - f(a)}{h} \end{talign} \end{block} \pause \begin{block}{} The slope $m$ is also called the \emph{slope of the curve} at the point. \end{block} \pause\medskip \begin{exampleblock}{} Find an equation of the tangent to $f(x) = \frac{3}{x}$ at point $(3,1)$. \pause\medskip The slope is: \begin{talign} m &= \lim_{h\to 0} \frac{f(3+h) - f(3)}{h} \mpause[1]{= \lim_{h\to 0} \frac{\frac{3}{3+h} - 1}{h}} \mpause[2]{= \lim_{h\to 0} \frac{\frac{3 - (3+h)}{3+h}}{h}}\\ &\mpause[3]{= \lim_{h\to 0} \frac{-h}{h(3+h)}} \mpause[4]{= \lim_{h\to 0} -\frac{1}{3+h}} \mpause[5]{= -\frac{1}{3}} \end{talign} \pause[10] Thus $y - 1 = -\frac{1}{3} (x-3)$\pause, that is, the tangent is $y = 2 - \frac{x}{3}$. \end{exampleblock} \end{frame}