If f:R→R is defined in f(x)=cos 3x-cos xx2 for ≠0λ for x=0 and if f is continuous at x = 0 , then |λ|=

λ=limx→0 cos 3x-cos xx2=limx→0 -3sin 3x+sin x2x =limx→0 -9cos 3x+cos x2=-9+12=-82=-4 ⇒ |-λ|=4

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If $f : R \to R$ is defined in $f (x) = \{\begin{matrix} \frac{\cos 3 x - \cos x}{x^{2}} & for \neq 0 \\ λ & for x = 0 \end{matrix}$ and if f is continuous at x = 0 , then $| λ | =$

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answer is 4.

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Detailed Solution

$\begin{array}{lc} λ = \lim_{x \to 0} \frac{\cos 3 x - \cos x}{x^{2}} = \lim_{x \to 0} \frac{- 3 \sin 3 x + \sin x}{2 x} = \lim_{x \to 0} \frac{- 9 \cos 3 x + \cos x}{2} = \frac{- 9 + 1}{2} = \frac{- 8}{2} = - 4 \\ \Rightarrow | - λ | = 4 \end{array}$