The point $P(9,5)$ lies on the curve $y=\sqrt[\phantom{2}]{x}+2.$ If $Q$ is the point $(x,\sqrt[\phantom{2}]{x}+2),$ find the slope of the secant line $PQ$ for the following values of $x.$

If $x=9\mathrm{.}1,$ the slope of $PQ$ is: $$ and if $x=9\mathrm{.}01,$ the slope of $PQ$ is: $$ and if $x=8\mathrm{.}9,$ the slope of $PQ$ is: $$ and if $x=8\mathrm{.}99,$ the slope of $PQ$ is: $$ Based on the above results, guess the slope of the tangent line to the curve at $P(9,5).$