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Given the following,
$$\displaystyle{f(x) = \left\{ \begin{array}{ccc} \sqrt{16-x^2} &,& x \lt 4\\ x^2-10x+24 &,& 4 \lt x \lt 7\\ 4 &,& x \ge 7 \end{array} \right.}$$Sketch $y=f(x)$, and then find each of the following: 
The domain of $f$
The range of $f$
All $x$ and $y$ intercepts
$f\,(0)$
$f\,(7)$
$f\,(10)$
$f\,(5)$
$\displaystyle{\lim_{x \rightarrow 4} \ f\,(x)}$
$\displaystyle{\lim_{x \rightarrow 7} \ f\,(x)}$
$\displaystyle{\lim_{x \rightarrow -4} \ f\,(x)}$
Evaluate each limit given below, provided that it exists, and provide a graphical interpretation (hole, gap, vertical asymptote, point of continuity, etc...) for each related function at the $x$-value in question.
$\displaystyle{\lim_{x \rightarrow 1} \; (x^2-2)}$ 
$\displaystyle{\lim_{x \rightarrow -3} \; \frac{x^2 - 9}{x+3}}$ 
$\displaystyle{\lim_{x \rightarrow 2} \; \frac{x^2-4}{|x-2|}}$ 
$\displaystyle{\lim_{x \rightarrow -1} \; \frac{1}{(x+1)^2}}$ 
$\displaystyle{\lim_{x \rightarrow \pi/2} \; \cos^2(2x)}$ 
$\displaystyle{\lim_{x \rightarrow 3} \; \frac{x-3}{x^2-9}}$ 
$\displaystyle{\lim_{x \rightarrow -3} \; \frac{x-3}{x^2-9}}$ 
$\displaystyle{\lim_{x \rightarrow 4} \; \frac{2-\sqrt{8-x}}{x-4}}$ 
$\displaystyle{\lim_{x \rightarrow 0} \; \frac{\sin 3x}{x}}$ 
$\displaystyle{\lim_{x \rightarrow 0} \; \frac{1-\cos x}{3x}}$ 
$\displaystyle{\lim_{x \rightarrow 2} \; (x^2 - 3x + 1)}$ 
$\displaystyle{\lim_{x \rightarrow -4} \; \frac{x^2-16}{x-4}}$ 
$\displaystyle{\lim_{x \rightarrow -4} \; \frac{x^2+3x-4}{x+4}}$ 
$\displaystyle{\lim_{x \rightarrow -3} \; \frac{x^2-9}{|x+3|}}$ 
$\displaystyle{\lim_{x \rightarrow 2} \; \frac{x^2+4}{x-2}}$ 
$\displaystyle{\lim_{x \rightarrow 3} \; \frac{x-5}{|x-3|}}$ 
Evaluate the given limit(s) provided they exist, and describe what happens in the graph of the function (hole, gap, vertical asymptote, point of continuity, etc...) at the related $x$-value(s).
$\displaystyle{\lim_{x \rightarrow 0} \; f(x)}$; $\displaystyle{f(x) = \left\{ \begin{array}{ccl}
x+2 &,& x \lt 0\\
0 &,& x=0\\
2 &,& x \gt 0\\
\end{array} \right.}$ 
$\displaystyle{\lim_{x \rightarrow 4} \; f(x)}$; $\displaystyle{f(x) = \left\{ \begin{array}{ccl}
x+2 &,& x \neq 4\\
2 &,& x = 4\\
\end{array}\right.}$ 
$\displaystyle{\lim_{x \rightarrow -2} \; f(x)}$; $\displaystyle{f(x) = \left\{ \begin{array}{ccl}
5x+7 &,& x \lt -2\\
x^2 &,& x \gt -2\\
\end{array}\right.}$ 
$\displaystyle{\lim_{x \rightarrow 0} \; f\,(x)}$; $\displaystyle{f(x) = \left\{ \begin{array}{ccc}
x^2 &,& x \lt 0\\
1+x &,& x \gt 0\\
\end{array}\right.}$ 
$\displaystyle{\lim_{x \rightarrow 0} \; f(x)}$, $\displaystyle{\lim_{x \rightarrow -1} \; f(x)}$; $\displaystyle{f(x) = \left\{ \begin{array}{ccl}
x+1 &,& x \lt 0\\
e^x &,& x \ge 0\\
\end{array} \right.}$ 
Evaluate the following limits, if they exist, and provide a graphical interpretation (hole, gap, vertical asymptote, point of continuity, etc...) of each related function.
$\displaystyle{\lim_{x \rightarrow 2} \; \frac{x^2-4}{x-2}}$ 
$\displaystyle{\lim_{x \rightarrow 0} \; \frac{\sqrt{x+3}-2}{x^2}}$ 
$\displaystyle{\lim_{x \rightarrow 1} \; \frac{x+1}{x^2-1}}$ 
$\displaystyle{\lim_{x \rightarrow 2} \; \frac{x-2}{x-2}}$ 
$\displaystyle{\lim_{x \rightarrow 4} \; \frac{1/\sqrt{x} - 1/2}{x^2-16}}$ 
$\displaystyle{\lim_{x \rightarrow 0} \; \frac{ 2 / (3x+1) - 3}{2x+3}}$ 
$\displaystyle{\lim_{x \rightarrow 8} \; \frac{\sqrt{x+1} + 2}{10}}$ 
$\displaystyle{\lim_{x \rightarrow 1} \; \frac{1 - 1/x^3}{x-1}}$ 
$\displaystyle{\lim_{x \rightarrow 7} \; \frac{3 - \sqrt{2x-5}}{x-7}}$ 
$\displaystyle{\lim_{x \rightarrow 2^{-}} \; \frac{x-2}{|x-2|}}$ 
$\displaystyle{\lim_{x \rightarrow 2} \; \frac{x^2-5x+6}{x^2-4x+4}}$ 
$\displaystyle{\lim_{x \rightarrow 5} \; \frac{x-5}{|x-3|}}$ 
$\displaystyle{\lim_{x \rightarrow 0} \; \frac{\sqrt{1+x} - 1}{2x}}$ 
$\displaystyle{\lim_{x \rightarrow 1} \; \sqrt{1-x}}$ 
$\displaystyle{\lim_{x \rightarrow 1} \; \frac{x^3 - x^2 - 2x + 2}{x^2 + x - 2}}$ 
$\displaystyle{\lim_{x \rightarrow 0} \left[ x \left( \frac{4}{x} -1 \right) \right]}$ 
Sketch the given function, and then find the indicated values.
$\displaystyle{g(x) = \left\{ \begin{array}{ccc}
10 &,& x \le -3\\
1-3x &,& -3 \lt x \lt 1\\
0 &,& x=1\\
x^2-3 &,& x \gt 1\\
\end{array}\right.}$ 
i) $\displaystyle{\lim_{x \rightarrow -4} \; g(x)}$ ii) $\displaystyle{\lim_{x \rightarrow -3} \; g(x)}$ iii) $\displaystyle{\lim_{x \rightarrow 0} \; g(x)}$ iv) $\displaystyle{\lim_{x \rightarrow 1} \; g(x)}$
$\displaystyle{f(x) = \left\{ \begin{array}{ccc}
-\sqrt{16-x^2} &,& x \lt 0\\
3 &,& x = 0\\
x^2 - 4 &,& 0 \lt x \le 3\\
3 + \sqrt{7-x} &,& x \gt 3\\
\end{array}\right.}$ 
i) $\displaystyle{\lim_{x \rightarrow -4} \; f(x)}$ ii) $\displaystyle{\lim_{x \rightarrow 0} \; f(x)}$ iii) $\displaystyle{\lim_{x \rightarrow 2} \; f(x)}$ iv) $\displaystyle{\lim_{x \rightarrow 3} \; f(x)}$ v) $f(0)$
$\displaystyle{f(x) = \left\{ \begin{array}{ccc}
\sqrt{49-x^2} &,& x \le 0\\
7 &,& 0 \lt x \le 3\\
x^2-8x+12 &,& x > 3\\
\end{array}\right.}$ 
i) $\displaystyle{\lim_{x \rightarrow 0} \; f(x)}$ ii) $\displaystyle{\lim_{x \rightarrow 1} \; f(x)}$ iii) $\displaystyle{\lim_{x \rightarrow 3} \; f(x)}$ iv) $\displaystyle{\lim_{x \rightarrow 4} \; f(x)}$
$\displaystyle{f(x) = \left\{ \begin{array}{ccc}
\sqrt{9-x^2} &,& x \lt 0\\
3-x &,& 0 \le x \lt 3\\
9-x^2 &,& x \gt 3
\end{array}\right.}$ 
i) $\displaystyle{\lim_{x \rightarrow 0} \; f(x)}$ ii) $\displaystyle{\lim_{x \rightarrow 2} \; f(x)}$ iii) $\displaystyle{\lim_{x \rightarrow 3} \; f(x)}$
Evaluate the following limits and give a graphical interpretation of the behavior of the related function at the indicated $x$ value. When a limit below fails to exist, cite the reason why this is the case.
$\displaystyle{\lim_{x \rightarrow -6} \sqrt{4-2x}}$ 
$\displaystyle{\lim_{x \rightarrow 6} \frac{x^2-6x}{x^2-7x+6}}$ 
$\displaystyle{\lim_{x \rightarrow -4} \frac{x^2-16}{|4+x|}}$ 
$\displaystyle{\lim_{x \rightarrow 4} \frac{\sqrt{25-x^2}-3}{4-x}}$ 
$\displaystyle{\lim_{x \rightarrow -1} \frac{|x+1|}{x+1}}$ 
$\displaystyle{\lim_{x \rightarrow 0} \frac{\sin 4x}{2x}}$ 
$\displaystyle{\lim_{x \rightarrow 3} \frac{|x-3|}{x^2-9}}$
$\displaystyle{\lim_{x \rightarrow -2} \frac{1-\sqrt{x^2-3}}{4-x^2}}$
$\displaystyle{\lim_{x \rightarrow \pi} \tan x}$
$\displaystyle{\lim_{x \rightarrow 1} \frac{\frac{1}{x} - 1}{x-1}}$
$\displaystyle{\lim_{x \rightarrow 0} \frac{e^{2x}}{\ln(x+e)}}$
$\displaystyle{\lim_{x \rightarrow -3} \frac{x^2-4x+4}{x^2+x-6}}$
$\displaystyle{\lim_{x \rightarrow 1} \frac{x^3-8}{x-2}}$
$\displaystyle{\lim_{x \rightarrow -3} \frac{9-x^2}{x+3}}$
$\displaystyle{\lim_{x \rightarrow 5} \frac{x-5}{x+2}}$
$\displaystyle{\lim_{x \rightarrow \pi/4} \tan(2x)}$
$\displaystyle{\lim_{x \rightarrow 1} \textrm{ Arccos } \left( \frac{1}{x} \right)}$
$\displaystyle{\lim_{x \rightarrow 0} \frac{x}{1-\cos x}}$
$\displaystyle{\lim_{x \rightarrow -2} \frac{x^2-4}{x^3+8}}$
$\displaystyle{\lim_{x \rightarrow 1} \frac{\sqrt{2x+1} - \sqrt{3}}{x-1}}$
$\displaystyle{\lim_{x \rightarrow 3^{-}} \frac{|x-3|}{x-3}}$
$\displaystyle{\lim_{x \rightarrow 0} \frac{x}{e^x}}$
Evaluate the following limits and give a graphical interpretation of the behavior of the related function at the indicated $x$ value. When a limit below fails to exist, cite the reason why this is the case.
$\displaystyle{\lim_{x \rightarrow 0} \ e^{3x} \ln |1-x|}$
$\displaystyle{\lim_{x \rightarrow 2} \ \textrm{ Arcsin } \left( \frac{1}{x} \right)}$
$\displaystyle{\lim_{x \rightarrow \pi} \ \sec \left( \frac{x}{2} \right)}$
$\displaystyle{\lim_{x \rightarrow 4} \ \frac{\sqrt{25-x^2} - 3}{4-x}}$
$\displaystyle{\lim_{x \rightarrow -2} \ \frac{\frac{16}{x^4} - 1}{x^2-4}}$
$\displaystyle{\lim_{x \rightarrow 4} \ \frac{|x-4|}{x^2-16} }$
$\displaystyle{\lim_{x \rightarrow 0} \ \frac{\tan(2x)}{x}}$
$\displaystyle{\lim_{x \rightarrow 2^-} \ \tan \left(\frac{\pi}{x} \right)}$
$\displaystyle{\lim_{x \rightarrow 5} \ \frac{x^3-125}{x^2-25}}$
$\displaystyle{\lim_{x \rightarrow 0} \ \textrm{Arctan} (2x)}$
$\displaystyle{\lim_{x \rightarrow 0} \ \sin \left( \frac{1}{2x} \right)}$
$\displaystyle{\lim_{x \rightarrow 2} \ \textrm{Arctan} \left( \frac{x}{2} \right)}$
$\displaystyle{\lim_{x \rightarrow -2} \ \frac{|x+2|}{x^3+8}}$
$\displaystyle{\lim_{x \rightarrow -1} \ \textrm{Arcsin } \left( \frac{x}{2} \right)}$
$\displaystyle{\lim_{x \rightarrow 1} \ \frac{\frac{1}{x} - 1}{x^3 - 1}}$
$\displaystyle{\lim_{x \rightarrow -3} \ \textrm{Arctan}^2 \left( \frac{x}{3} \right)}$
$\displaystyle{\lim_{x \rightarrow 0} \ \sin \left( \frac{4}{3x} \right)}$
$\displaystyle{\lim_{x \rightarrow 4} \ \frac{|x-4|}{x^2-16}}$
$\displaystyle{\lim_{x \rightarrow -2} \ \frac{1-\sqrt{x^2-3}}{4-x^2}}$