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Compute the following limits$.$

$($a$)\underset{x3}{lim}\frac{x^3-2x^2-9}{x^2-2x-3}$

$($b$)\underset{x4}{lim}\frac{4x-x^2}{2-\sqrt[\phantom{2}]{x}}$

Let $f(x)=\frac{1}{x}$ and define the function

$g(x)=\underset{x0}{lim}\frac{f(xx)-f(x)}{x}$

$($a$)$ Calculate $g(-1),g(1)$ and $g(3)$ and plot the points $(-1,g(-1)),(3,g(3))$ and $(1,g(1)).$

$($b$)$ Give a function for $g(x)$ that has no limits and plot that function. Hint, treat $x$ as a constant and evaluate the limit in the $g(x)$ function.