\begin{frame} \frametitle{Computing Limits: Direct Substitution Property} \begin{block}{Direct Substitution Property} If $f$ is a polynomial or a rational and $a$ is in the domain of $f$, then: \begin{talign} \lim_{x\to a} f(x) = f(a) \end{talign} \end{block} \pause Works also for one-sided limits $\lim_{x\to a^{\pm}} f(x) = f(a)$.\\ \pause Works also for algebraic functions if $f(x)$ is defined close to $a$. % Functions for which $\lim_{x\to a} f(x) = f(a)$ are called continuous at $a$.\hspace*{-1ex} % (we will study them in one of the following lectures) \pause\bigskip \begin{exampleblock}{} The function $f(x) = 2x^2 - 3x + 4$ \pause is a polynomial and hence:\pause \begin{talign} \lim_{x \to 5} f(x) \mpause[1]{= f(5)} \mpause[2]{= 2\cdot 5^2 - 3\cdot 5 + 4 = 39} \end{talign} \end{exampleblock} \pause\pause\pause\medskip \begin{exampleblock}{} The function $g(x) = \frac{x^3 + 2x^2 -1}{5-3x}$ \pause is rational and $-2$ is in the domain; hence:\pause \begin{talign} \lim_{x \to -2} g(x) \mpause[1]{= g(-2) }\mpause[2]{= \frac{(-2)^3 + 2\cdot (-2)^2 - 1}{5-3\cdot (-2)} = -\frac{1}{11}} \end{talign} \end{exampleblock} \end{frame}