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\problemgrouptitle{One-sided Limits}

\item Find the one-sided limits of $f$ at the values $x=0$, $x=2$ and $x=4$.

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\answer{$\displaystyle\lim_{x\to0^-}f=4$, $\displaystyle\lim_{x\to0^+}f=4$, $\displaystyle\lim_{x\to2^-}f=3$, $\displaystyle\lim_{x\to2^+}f=2$, $\displaystyle\lim_{x\to4^-}f=\infty$ (limit does not exist but approaches infinity), $\displaystyle\lim_{x\to4^+}f=2$}

\item Find the one-sided and two-sided limits of $f$ at the values $x=0$, $x=2$ and $x=4$.
	
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\answer{$\displaystyle\lim_{x\to0^-}f=-1$, $\displaystyle\lim_{x\to0^+}f=-1$, $\displaystyle\lim_{x\to2^-}f=1$, $\displaystyle\lim_{x\to2^+}f=-2$, $\displaystyle\lim_{x\to4^-}f=2$, $\displaystyle\lim_{x\to4^+}f=2$, $\displaystyle\lim_{x\to0}f=-1$, $\displaystyle\lim_{x\to2}f$ does not exist, $\displaystyle\lim_{x\to4}f=2$}

\item Suppose $\displaystyle f(x)=\left\{
\begin{array}{cc}
\frac{4}{x+4} & \mbox{if}\ x<2 \\
x^2+1 & \mbox{if}\ x\ge2
\end{array}
\right.$\ .

Find $\displaystyle\lim_{x\to 2^-}f(x)$, $\displaystyle\lim_{x\to 2^+}f(x)$, and $\displaystyle\lim_{x\to 2}f(x)$, if they exist.

\answer{$\displaystyle\lim_{x\to 2^-}f(x)=\frac{2}{3}$, $\displaystyle\lim_{x\to 2^+}f(x)=5$, $\displaystyle\lim_{x\to 2}f(x)$ does not exist as the left and right-handed limits are not equal at $x=2$\,.}

\item Suppose $\displaystyle f(x)=\left\{
\begin{array}{cc}
x^2+2cx & \mbox{if}\ x\le-1 \\
x+5c & \mbox{if}\ x>-1
\end{array}
\right.$\ .

Find all values for the constant $c$ that make the two-sided limit exist at $x=-1$\,.
\answer{$\displaystyle c=\frac{2}{7}$}

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