$(1)$ Determine if the function is continuous at $x=0.$ If it is not, determine if it is continuous from the left, continuous from the right, or neither.

$f(x)=\{\begin{array}{cc}x+e^x,& x<mn>0</mn>\\ x^2,& x0\end{array}$

$(2)$ Find the value of $k$ that makes the function continuous.

$f(x)=\{\begin{array}{cc}\frac{1}{2}x-2,& x1\\ -x+k,& x<mn>1</mn>\end{array}$

$(3)$ Use the intermediate value theorem to determine if the equation has at least one solution in the specified interval.

$x^2-4ln(5x+2)=0,on0,4$

$(4)\underset{x}{lim}ln(\frac{10x+8x^2}{x^2-1})$

$(5)$ Find all a symptotes

$($a$)f(x)=\frac{4x+10}{x^2-2x-15}$

$($b$)f(x)=\frac{3x^2+1}{5-10x^2}$

$(6)$ Find the equation of the tangent line to $f(x)=\frac{5}{x}$ at $x=10.$

$(7)$ Use the definition of a derivative to find the derivative of:

$($a$)f(x)=x^3-2x^2$

$($b$)f(x)=\sqrt[\phantom{2}]{3x-4}$

$($c$)f(x)=\frac{2x+1}{x+5}$