Find the value of the constant `k` so that the function given below is continuous at `x=0` . `f(x)={(1-cos2x)/(2x^2),\ \ \ x!=0\ \ \ \ \ \ \ \ k ,\ \ \ \ \ \ \ \ \ \ \ \ \ x=0`

Explanation

`lim _{x rightarrow 0-} frac{1-cos 2 x}{2 x^{2}}`

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=lim _{x rightarrow 0^{-}} frac{1-(1-2 sin ^{2} x)}{2^{2}}`

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=lim _{x rightarrow 0^{-}} frac{2 sin ^{2} x}{2 x^{2}}`

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=lim _{x rightarrow 0^{-}} frac{sin ^{2} x}{x^{2}}`

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=1`

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And lim _{x rightarrow 0^{-}} frac{1-cos 2 x}{2 x^{2}}`

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=lim _{x rightarrow 0^{-}} frac{1-(1-2 sin ^{2} x)}{2 x^{2}}`

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=lim _{x rightarrow 0^{-}} frac{2 sin ^{2} x}{2 x^{2}}`

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