The value of $ \lim _{\mathrm{x} \rightarrow 0}\left(\left[\frac{100 \mathrm{x}}{\sin \mathrm{x}}\right]+\left[\frac{99 \sin \mathrm{x}}{\mathrm{x}}\right]\right) $, (where $ [\mathrm{x}] $ represents greatest integral function less than or equal to $ \mathrm{x} $ ) is $ 99 \lambda . $ Then the value of $ \lambda $ is

Question

The value of $ \lim _{\mathrm{x} \rightarrow 0}\left(\left[\frac{100 \mathrm{x}}{\sin \mathrm{x}}\right]+\left[\frac{99 \sin \mathrm{x}}{\mathrm{x}}\right]\right) $, (where $ [\mathrm{x}] $ represents greatest integral function less than or equal to $ \mathrm{x} $ ) is $ 99 \lambda . $ Then the value of $ \lambda $ is,

MARKS App · Accepted Answer

We know that, lim x → 0 sin x x → 1 - then, lim x → 0 100 x sin x ⇒ lim x → 0 100 sin x x = 100 lim x → 0 99 sin x x ⇒ lim x → 0 99 sin x x = 98 Hence ⇒ lim x → 0 [ 100 x sin x ] + [ 99 sin x x ] = 100 + 98 = 198 9 9 λ = 198 Hence, λ = 2

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