Section $2\mathrm{.}5-$ Continuity

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval $ln(x)=x-\sqrt[\phantom{2}]{x},(2,3)$

Section $2\mathrm{.}6-$ Limits at Infinity; Horizontal Asymptotes

$**5.$ Find the limit or show that it does not exist.

$\underset{x-}{lim}\frac{x-2}{x^2+1}$

Section $2\mathrm{.}7-$ Derivatives and Rates of Change

$6.$ Find an equation of the tangent line to the curve at the given point.

$y=x^3-3x+1,(2,3)$

Section $2\mathrm{.}8-$ The Derivative as a Function

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$f(x)=\sqrt[\phantom{2}]{9-x}$

$2$