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\problemgrouptitle{Limits Review}

\begin{problemblock}{Evaluate the limits.}

\addtoanswerkey{\begin{multicols}{5}}

\item $\displaystyle \lim_{x\to 3}\frac{2x^2-5x-3}{x^2+x-12}$
\answer{$1$}

\item $\displaystyle \lim_{t\to 4}\frac{\sqrt{t}-2}{t^2-2t-8}$
\answer{$\displaystyle \frac{1}{24}$}

\item $\displaystyle \lim_{x\to -1}\frac{x^3+x^2+2x+2}{x^2-2x-3}$
\answer{$\displaystyle -\frac{3}{4}$}

\item $\displaystyle \lim_{t\to -2}\frac{\sqrt{10+3t}-2}{3t^2+4t-4}$
\answer{$\displaystyle -\frac{3}{32}$}

\item $\displaystyle \lim_{t\to 2}\frac{\frac{8}{t}-4}{3t^2-4t-4}$
\answer{$\displaystyle -\frac{1}{4}$}

\addtoanswerkey{\end{multicols}}

\end{problemblock}

\begin{problemblock}{Evaluate the trigonometric limits.}

\addtoanswerkey{\begin{multicols}{5}}

\item $\displaystyle \lim_{x\to 0}\frac{\sin(4x)}{\sin(5x)}$
\answer{$\displaystyle \frac{4}{5}$}

\item $\displaystyle \lim_{\theta\to 0}\frac{\sin(3\theta)}{\tan(4\theta)}$
\answer{$\displaystyle \frac{3}{4}$}

\item $\displaystyle \lim_{t\to \pi}\frac{2\sin^2t}{1+\cos t}$
\answer{$4$}

\item $\displaystyle \lim_{x\to 0}\frac{\cos(3x)+\cos(4x)-2}{x}$
\answer{$0$}

\addtoanswerkey{\end{multicols}}

\end{problemblock}

\begin{problemblock}{Determine whether the functions are continuous at the given value.}

\addtoanswerkey{\begin{multicols}{3}}

\item $\displaystyle f(x)=\frac{x+3}{\sqrt{x^2+5}}$ at $x=-1$
\answer{Continuous}

\item $\displaystyle h(t)=\frac{t^2+2t-1}{t-3}$ at $t=3$
\answer{Not continuous}

\item $\displaystyle g(x)=\left\{\begin{array}{c}
3x^2-1,\ \mbox{if\ }x\leq 2 \\ 
\frac{x^2+x-6}{x-2},\ \mbox{if\ }x>2
\end{array}\right.$ at $x=2$
\answer{Not continuous}

\addtoanswerkey{\end{multicols}}

\end{problemblock}
