Evaluate the limit L= _ x 0 x- x (1+x)-x+ x^ 2 2 .

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

Evaluate the limit

L=\lim_{x\to 0}\frac{x-\sin x}{\ln(1+x)-x+\dfrac{x^{2}}{2}}.

(a) Show that direct substitution results in

0/0

. (b) Calculate the exact value of

L

and justify each step in your calculation.