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The value of f 0 so that the function f x = 1 - cos 1 - cos x x 4 is continuous everywhere is k , then value of 10 k is

Correct answer

1.25

Step-by-step solution

If f x is continuous at x = 0 f x = lim x → 0 f x = lim x → 0 1 - cos 1 - cos x x 4 × 1 + cos 1 - cos x 1 + cos 1 - cos x = lim x → 0 sin 2 1 - cos x x 4 · 1 + cos 1 - cos x · 1 - cos x 2 1 - cos x 2 = lim x → 0 sin 1 - cos x 1 - cos x 2 × lim x → 0 1 - cos x x 2 2 × lim x → 0 1 1 + cos 1 - cos x = 1 2 × 1 4 × 1 2 = 1 8 ∴ k = 1 8 ⇒ 10 k = 5 4 = 1.25

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