Problem Statement: Let be differentiable when . Given that and for every show that (a)$ latex f(x)$ is differentiable on and (b) evaluate .
Proof: First consider . We know this limit exists since is differentiable with . Furthermore, since we may conclude that
Let . In order for to be differentiable at we must show that exists.
Since is defined for every it follows that is differentiable on and that for every . Since we are given that is differentiable at it follows that is differentiable on
Reflection: This one wasn’t too bad. The ball dropped when I realized that was a fixed number and not changing. The main idea here is that since we know we can factor out that from both terms and end up with a limit we already know about.