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UPSC NDA I 2022 Mathematics Solved Paper

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Question : 74 of 120

Marks: +1, -0

What is

\lim\limits_{x \to 0} x^3 (\csc x)^2

equal to ?

0
$\frac{1}{2}$
1
Limits does not exist

Solution:

Concept:

Using the standard limit

\lim\limits_{x \to 0} \frac{\sin x}{x} = 1

to simplify the expression.

Explanation:

Given:

\lim\limits_{x \to 0} x^3 (\csc x)^2

Since

\csc x = \frac{1}{\sin x}

, rewrite as:

\lim\limits_{x \to 0} \frac{x^3}{(\sin x)^2}

Factor as:

\lim\limits_{x \to 0} \left( \frac{x}{\sin x} \right)^3 \cdot \sin x

We know

\lim\limits_{x \to 0} \frac{\sin x}{x} = 1

, so

\lim\limits_{x \to 0} \frac{x}{\sin x} = 1

Thus the limit becomes:

\lim\limits_{x \to 0} (1)^3 \cdot \sin x = \lim\limits_{x \to 0} \sin x

As

x \to 0

,

\sin x \to 0

Therefore the limit equals

0

Answer:

0

(Option A)

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