The function is defined for .

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

The function

g(x)=\dfrac{1-\cos x}{x^{2}}

is defined for

x\neq 0

. (a) Prove that

0\le 1-\cos x\le \dfrac{x^{2}}{2}

for all real

x

. Clearly indicate any theorems or geometric facts you invoke. (b) Using the result from part (a), evaluate

\displaystyle\lim_{x\to 0} \dfrac{1-\cos x}{x^{2}}

. Show each step of your argument. (c) Determine whether

g

can be continuously extended at

x=0

. If so, define

g(0)

and justify continuity at that point.