Question

The value of $f(0)$ so that the function $f (x) = \frac{1-\cos\left(1-\cos\,x\right)}{x^{4}}$ is continuous everywhere is

• A. $\frac{1}{2}$
• B. $\frac{1}{4}$
• C. $\frac{1}{6}$
• D. $\frac{1}{8}$

Answer 1

Answer: (D)

$\displaystyle\lim _{x \rightarrow 0} \frac{1-\cos (1-\cos x)}{x^{4}}$
$=\displaystyle\lim _{x \rightarrow 0} \frac{2 \sin ^{2}\left(\frac{2 \sin ^{2}\left(\frac{x}{2}\right)}{2}\right)}{x^{4}}$
$=2 \displaystyle\lim _{x \rightarrow 0} \frac{\sin ^{2}\left(\sin ^{2}\left(\frac{x}{2}\right)\right)\left(\sin ^{2}\left(\frac{x}{2}\right)\right)^{2}}{x^{4}\left(\sin ^{2}\left(\frac{x}{2}\right)\right)^{2}}$
$=2 \displaystyle\lim _{x \rightarrow 0} \frac{\sin ^{4}\left(\frac{x}{2}\right)}{\left(\frac{x}{2}\right)^{4} 2^{4}}=\frac{1}{2^{3}}=\frac{1}{8}$