Problem Statement: Use the definition of the derivative to show that if and if is differentiable at .
Proof: By definition
To evaluate this limit we will apply the Squeeze Theorem. Since for all it follows that as and that as . Either way,
So, by the Squeeze Theorem it follows that . Thus, is differentiable at since exists and is finite.
Reflection: This one was very straight forward. When I first did the problem I didn’t go through the Squeeze Theorem argument though. After re-reading the proof and trying to defend it to Jeremy I realized that the Squeeze Theorem argument is kind of necessary. I told him, “well, on the qual I would write that out” and he reminded me of what my high school volleyball coach constantly told our team, “you practice how you play.” So, in the future I will be making sure to write out every single detail every time I take a practice exam.
On another note, I’ve been timing these exams and I am having enough time, yay!