$[-/1$ Points$]$

SCALCET$92\mathrm{.}1\mathrm{.}003.$

The point $P(4,-4)$ lies on the curve $y=\frac{4}{3-x}.$

$($a$)$ If $Q$ is the point $(x,\frac{4}{3-x}),$ find the slope of the secant line $PQ($correct to six decimal places$)$ for the following values of $x.$

$($i$)3\mathrm{.}9$

$m_{PQ}=$

$$

$($ii$)3\mathrm{.}99$

$m_{PQ}=$

$$

$($iii$)3\mathrm{.}999$

$m_{PQ}=$

$$

$($iv$)3\mathrm{.}9999$

$m_{PQ}=$

$$

$($v$)4\mathrm{.}1$

$m_{PQ}=$

$$

$($vi$)4\mathrm{.}01$

$m_{PQ}=$

$$

$($vii$)4\mathrm{.}001$

$m_{PQ}=$

$$

$($viii$)4,0001$

$m_{PQ}=$

$$

$($b$)$ Using the results of part $($a$),$ guess the value of the slope of the tangent line to the curve at $P(4,-4).$

$m=$

$$

$($c$)$ Using the slope from part $($b$),$ find an equation of the tangent line to the curve at $P(4,-4).$