Applications of Differential Calculus to Limit Calculations and Extremum Problems

Abstract

Certain kinds of limit computations, those involving the so-called indeterminate forms, can be accomplished using two rules explained in this chapter, L’Hopital’s rule and a rule based on the Taylor–Maclaurin series. Again, studying the Taylor series around critical points gives sufficient conditions for the presence of a local (relative) extremum. Furthermore, whether the curve is bending (upward or downward) depends on the sign of higher order derivatives. Necessary and sufficient conditions for existence of an asymptote to a curve are proved. Finally, such investigations are used in curve sketching.