I think I can do it without what you said but using the ideas from calculus. Your limit represent the slope of the tangent of at . Let
.
The tangent line is or
The intersection of the tangent line and the quintic
or
There should be two intersection points, at and (some ) where the first will be a double root so eqn. (1.1) has the form
for some and .
Expanding and equating the coefficients of and gives
If we add/substract in the following order
gives
or , our desired limit.