at $3$:$20$ PM by averaging the slopes of two secant lines.

$3.$ The point $P(2,-1)$ lies on the curve $y-\frac{1}{1-x}.$

$($a$)$ If $Q$ is the point $(x,\frac{1}{1-x}),$ find the slope of the secant line PQ $($correct to six decimal places$)$ for the following values of $x$ :

$($i$)1\mathrm{.}5$

$($ii$)1\mathrm{.}9$

$($iii$)1\mathrm{.}99$

$($iv$)1\mathrm{.}999$

$($v$)2\mathrm{.}5$

$($vi$)2\mathrm{.}1$

$($vii$)2\mathrm{.}01$

$($viii$)2\mathrm{.}001$

$($b$)$ Using the results of part $($a$),$ guess the value of the slope of the tangent line to the curve at $P(2,-1).$

$($c$)$ Using the slope from part $($b$),$ find an equation of the tangent line to the curve at $P(2,-1).$