GATE Civil Engineering (CE) 2019 Shift 2 Solved Paper

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Question : 24 of 65

Marks: +1, -0

The following inequality is true for all

x

close to 0 .

2-\;{x^{2}}/{3}<\;{x \sin x}/{1-\cos x}<2

What is the value of

\lim ↙{x → 0} \;{x \sin x}/{1-\cos x} ?

$2$
$1 / 2$
$0$
$1$

Solution:

\lim ↙{x → 0} \;{x \sin x}/{1-\cos x}

?
It is

\;{0}/{0}

form
L's hospital rule (differentiate)

⇒ \;\;\lim↙{x→0} \;{x \cos x+\sin x}/{-(-\sin x)}

Putting 0 everywhere

\Rightarrow

still

\;{0}/{0}

form.
Differentiating numerator and denominator again,

\lim \;{x(-\sin x)+\cos x+\cos x}/{\cos x \atop x → 0}=2

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