The point $P(8,-3)$ lies on the curve $y=\frac{3}{7-x}.$

$($a$)$ If $Q$ is the point $),$ use your calculator to find the slope $m_{PQ}$ of the secant line $PQ($correct to six decimal places$)$ for the following values of $x.$

$(i)7\mathrm{.}9$

$m_{PQ}=$

$(ii)7\mathrm{.}99$

$m_{PQ}=$

$(iii)7\mathrm{.}999$

$m_{PQ}=$

$(iv)7\mathrm{.}9999$

$m_{PQ}=$

$(v)8\mathrm{.}1$

$m_{PQ}=$

$(vi)8\mathrm{.}01$

$m_{PQ}=$

$($vii$)8\mathrm{.}001$

$m_{PQ}=$

$($viii$)8\mathrm{.}0001$

$m_{PQ}=$

$($b$)$ Using the results of part $($a$),$ guess the value of the slope $m$ of the tangent line to the curve at $P(8,-3).$

$m=$

$($c$)$ Using the slope from part $($b$),$ find an equation of the tangent line to the curve at $P(8,-3).$