Consider the function defined by
Find values for the constants and
such that the derivative
exists.
We know that the derivative exists if and only if
exists. Furthermore, this limit exists if and only if the one-sided limits both exist and are equal:
So, plugging in the formula for (which is
if we approach
from the right, and is
if we approach from the left, and noting that
from the definition of
) we have,
For the limit on the left to exist we must have (otherwise the limit will diverge as
). Furthermore, this limit must be 0 since
is a constant (and the limit of
as
is 0). Therefore, we have
, and we have the equation
Therefore, and
are the values of the requested constants in terms of
.
how is it to use x tending to c+ and c- and f(c)