If limx→0 ⁡cos 2xcos 3x1x2=ek then the value of [k] is

elimx→0cos 2x-cos 3xx2⋅1cos 3x=ek e9-42(1)=ek⇒(k=5/2)⇒[2.5]=2

If ${\lim_{x \to 0}}_{} {(\frac{\cos 2 x}{\cos 3 x})}^{\frac{1}{x^{2}}} = e^{k}$ then the value of $[k]$ is

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answer is 2.

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Detailed Solution

$e^{\lim_{x \to 0} (\frac{\cos 2 x - \cos 3 x}{x^{2}} \cdot \frac{1}{\cos 3 x})} = e^{k} e^{\frac{9 - 4}{2}} (1) = e^{k} \Rightarrow (k = 5 / 2) \Rightarrow [2 .5] = 2$

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