The roots of the nominator are:
[tex] \frac{ 2 ( + - ) \sqrt{4 + 12}}{2} = \frac{ 2 ( + - ) 4}{2} = 3 \: and \: - 1[/tex]
Which means that the nominator can be written as:
[tex] {x}^{2} - 2x - 3 = (x - 3)(x + 1)[/tex]
Thus we compute the limit as shown below:
[tex]lim( \frac{ {x}^{2} - 2x - 3}{x - 3})(x - > 3) = lim( \frac{(x - 3)(x + 1)}{x - 3})(x - > 3) = lim( x + 1)(x - > 3) = 4[/tex]
