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Question

The value of $$f(0)$$ so that the function $$f(x)=\dfrac{1-cos(1-cos\,x)}{x^4}$$ is continuous every where is


A
12
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B
14
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C
16
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D
18
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Solution

The correct option is D $$\dfrac{1}{8}$$
$$\underset{x\rightarrow0}{lim}=\dfrac{1-cos(1-cos\,x)}{x^4}$$
$$=\underset{x\rightarrow0}{lim}\dfrac{2sin^2\begin{pmatrix}\dfrac{2sin^2\begin{pmatrix}\dfrac{x}{2}\end{pmatrix}}{2}\end{pmatrix}}{x^4}$$
$$=2\underset{x\rightarrow0}{lim}\dfrac{sin^2\begin{pmatrix}sin^2\begin{pmatrix}\dfrac{x}{2}\end{pmatrix}\end{pmatrix}\begin{pmatrix}sin^2\begin{pmatrix}\dfrac{x}{2}\end{pmatrix}\end{pmatrix}^2}{x^4\begin{pmatrix}sin^2\begin{pmatrix}\dfrac{x}{2}\end{pmatrix}\end{pmatrix}^2}$$
$$=2\underset{x\rightarrow0}{lim}\dfrac{sin^4\begin{pmatrix}\dfrac{x}{2}\end{pmatrix}}{\begin{pmatrix}\dfrac{x}{2}\end{pmatrix}^42^4}=\dfrac{1}{2^3}=\dfrac{1}{8}$$

Mathematics

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