Consider the function defined by
Find values for the constants and such that the derivative exists.
We know that exists if and only if
exists. This limit exists if and only if the two one-sided limits exits and are equal:
Using the definition of , we then evaluate these limits,
In simplifying the right hand side we used that . Furthermore, for the limit on the left to exist we must have (otherwise the limit will diverge as ). Now for the expression on the right, we claim the limit in the expression is 0. We can see this because
But this limit is the derivative of at . Since and , this term is indeed 0.
So, coming back to our original equations we then have,
Furthermore, since we already established that we have,
Therefore, the expressions for and we are asked to find are,