Review Derivatives BreakGround

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Two young mathematicians think about derivatives.

Check out this dialogue between two calculus students (based on a true story):

Devyn: Riley, I’ve got a weird limit I’m trying to evaluate.
Riley: What is it?
Devyn: I’m trying to find $\lim _{x \to 0} \frac {(e^x + \sin x) - 1}{x}.$
Riley: Have you tried just plugging in $0$ ?
Devyn: Yes, but it looks like a $\relax \boldsymbol {\tfrac {0}{0}}$ form. Usually I would just do some algebra to simplify the limit and it would work out, but there doesn’t seem to be any algebra to do.
Riley: Hmmm... I want to try something. What if we take that $e^x + \sin x$ group and call it $f(x)$ . Then this limit is $\lim _{x \to 0} \frac {f(x) - 1}{x}.$
Devyn: Oh! With that choice of $f$ , we know $f(0) = 1$ , so I can rewrite that limit as $\lim _{x \to 0} \frac {f(x) - f(0)}{x-0}.$
Riley: That looks familiar!

In trying to evaluate Devyn’s limit, Riley and Devyn managed to rewrite it into a form they had worked with before. Why does this limit look familiar?

It is a $\relax \boldsymbol {\tfrac {0}{0}}$ -form. It is the definition of the derivative. It is how you solve optimization problems. It becomes a Squeeze Theorem problem. None of the above.

Use what you found out about that form to find the value of the Devyn’s limit? $\displaystyle \lim _{x \to 0} \frac {(e^x + \sin x) - 1}{x} = \answer {2}$ .