Problem $1$: Finding Tangent Lines $(60$ points$)$

A piecewise defined function $g$ is given by the formula

$g(x)=\{\begin{array}{ccc}11-x^2& if& x-1\\ 5\sqrt[\phantom{2}]{2x+6}& if& x>-1\end{array}$

A partial graph of this function is given below. In this homework, we are interested in the tangent line to $g(x)$ at $x=-1.$ Work through the following parts to find a formula for this tangent line, or show it does not exist.

In this written homework, you will use the definition of the derivative to analyze a tricky derivative. Make sure you have practiced using the definition of the derivative to make calculations. Sample problems can be found in the Ximera textbook, Ximera homework, and recitation worksheets.

d$)1$ points$)$ First let's make a guess about what we will find. Do you think the tangent line to $g(x)$ at $x=-1$ will exist? Why or why not?

$[$Briefly explain your thinking. You may refer to the given graph in your explanations. You do not need to do any calculations for this part. This is a guess, and we are more interested in your thinking than in the exact numbers.$]$