$(9)$ The point $P(1,0)$ lies on the curve $y=sin(10\frac{}{x}).$

$($a$)$ If $Q$ is the point $(x,sin(10\frac{}{x})),$ find the slope of the secant line $PQ($correct to four decimal places$)$ for $x=2,1\mathrm{.}5,1\mathrm{.}4,1\mathrm{.}3,1\mathrm{.}2,1\mathrm{.}1,0\mathrm{.}5,0\mathrm{.}6,0\mathrm{.}7,0\mathrm{.}8,$ and $0\mathrm{.}9.$ Do the slopes appear to be approaching a limit$?$

$($b$)$ Use a graph of the curve to explain why the slopes of the secant lines in part $($a$)$ are not close to the slope of the tangent line at $P.$

$($c$)$ By choosing appropuiate secant lines, estimate the slope of the tangent line at $P.$