Question

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

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

$f\left(0\right)=RHL=\underset{x\to 0^{+}}{lim}f\left(x\right)=\underset{h\to 0}{lim}f\left(h\right)=\underset{h\to 0}{lim}\frac{1-cos(1-cosh)}{h^4}\times \frac{1+cos(1-cosh)}{1+cos(1-cosh)}=\underset{h\to 0}{lim}\frac{si{n}^2(1-cosh)}{h^4.(1+cos(1-cosh\left)\right)}.\underset{h\to 0}{lim}{\left(\frac{1-cosh}{h^2}\right)}^2\times \underset{h\to 0}{lim}\frac{1}{1+cos(1-cosh)}=(1{)}^2\times \frac{1}{4}\times \frac{1}{2}=\frac{1}{8}$