The value of $\lim _{x \rightarrow 0} 2\left(\frac{1-\cos x \sqrt{\cos 2 x} \sqrt[3]{\cos 3 x} \ldots \ldots \sqrt[10]{\cos 10 x}}{x^{2}}\right)$ is ______ .

JEE Main · 2024 · 08 Apr 2024 · Shift 1 · Limits

The value of

lim_{x \to 0} 2 (\frac{1 - c o s x c o s 2 x 3 c o s 3 x \dots\dots 10 c o s 10 x}{x ^{2}})

is ______ .

Answer: 55

Using

m cos m x \approx 1 - \frac{1}{m} \cdot \frac{( m x ) ^{2}}{2} = 1 - \frac{m x ^{2}}{2}

for small

x

, the product

cos x cos 2 x 3 cos 3 x \dots 10 cos 10 x \approx 1 - \frac{x ^{2}}{2} (1 + 2 + \dots + 10)

. So

\frac{1 - product}{x ^{2}} \to \frac{1}{2} (1 + 2 + \dots + 10) = \frac{1}{2} \cdot 55

, and multiplying by

2

gives

55

JEE Main · 2024 · 08 Apr 2024 · Shift 1 · Integer answer

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