Mathematics

Problem

Consider the limit

L=\lim_{x\to 0}\frac{1-\cos(4x)}{x^2}.

The expression appears to be indeterminate of the form

\tfrac00

. a. Verify algebraically that

1-\cos(4x)=2\sin^2(2x).

b. Rewrite

L

entirely in terms of

\sin(2x)

and

x

, showing each algebraic step clearly. c. Explain, without using L’Hôpital’s Rule, why the form obtained in part (b) guarantees that

L

approaches a finite, non-zero value as

x\to0

. State that value explicitly.