\begin{frame} \frametitle{Computing Limits: Function Replacement} Function replacement for one-sided limits: \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} \pause \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} \pause\bigskip \begin{exampleblock}{} Find $\lim_{x\to 2^-} g(x)$ where \begin{talign} g(x) = \begin{cases} x^2 &\text{for $x < 2$}\\ 5x+1 &\text{for $x \ge 2$} \end{cases} \end{talign} \pause We have \begin{talign} g(x) = x^2 \quad \text{for all $x < 2$} \end{talign} \pause Hence: \begin{talign} \lim_{x \to 2^-} g(x) = \lim_{x \to 2^-} x^2 \mpause[1]{ = 4} \end{talign} \end{exampleblock} \vspace{10cm} \end{frame}