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JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Continuity and Differentiability

If the function

f (x) = \frac{c o s ( s i n x ) - c o s x}{x ^{4}}

is continuous at each point in its domain and

f (0) = \frac{1}{k}

, then k is ____

Answer: 6

Solution

Continuity at

x = 0

forces

f (0) = lim_{x \to 0} f (x)

. With

cos A - cos B = 2 sin \frac{A + B}{2} sin \frac{B - A}{2}

, taking

A = sin x

and

B = x

,

cos (sin x) - cos x = 2 sin \frac{x + sin x}{2} sin \frac{x - sin x}{2} .

As

x \to 0

,

\frac{x + sin x}{2} \sim x

and

x - sin x \sim \frac{x ^{3}}{6}

, so

\frac{x - sin x}{2} \sim \frac{x ^{3}}{12}

. Since

sin u \sim u

for small

u

,

x \to 0 lim \frac{cos ( sin x ) - cos x}{x ^{4}} = x \to 0 lim \frac{2 \cdot x \cdot \frac{x ^{3}}{12}}{x ^{4}} = \frac{1}{6} .

So

f (0) = \frac{1}{6} = \frac{1}{k}

and

k = 6

.

JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Integer answer

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