\begin{frame} \frametitle{Computing Limits: Function Replacement} For one-sided limits we have: \begin{block}{} If $f(x) = g(x)$ for all $x < a$, then $\lim_{x \to a^-} f(x) = \lim_{x \to a^-} g(x)$. \end{block} \begin{block}{} If $f(x) = g(x)$ for all $x > a$, then $\lim_{x \to a^+} f(x) = \lim_{x \to a^+} g(x)$. \end{block} \bigskip \begin{exampleblock}{} Find $\lim_{x\to 0} |x|$ where \begin{talign} |x| = \begin{cases} x&\text{for $x \ge 0$}\\ -x&\text{for $x < 0$} \end{cases} \end{talign} \pause Since $|x| = x$ for all $x > 0$ we obtain: \begin{talign} \lim_{x\to 0^+} |x| = \lim_{x\to 0^+} x \mpause[1]{ = 0} \end{talign} \pause\pause Since $|x| = -x$ for all $x < 0$ we obtain: \begin{talign} \lim_{x\to 0^-} |x| = \lim_{x\to 0^-} -x \mpause[1]{ = 0} \end{talign} \pause\pause Hence $\lim_{x\to 0} |x| = 0$. \end{exampleblock} \vspace{10cm} \end{frame}