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UPSC NDA 2 2019 Mathematics Solved Paper

Question : 73 of 120

Marks: +1, -0

\underset{x→0}{lim} \frac{sinx∘}{tan3x∘}

Solution:

Concept:

Use the standard limits:

\displaystyle\lim_{\theta\to 0}\frac{\sin\theta}{\theta}=1

and

\displaystyle\lim_{\theta\to 0}\frac{\tan\theta}{\theta}=1

Explanation:

The given limit involves angles in degrees.

Convert degrees to radians:

x^\circ = \frac{\pi x}{180}

radians.

Thus,

\displaystyle\lim_{x\to 0}\frac{\sin x^\circ}{\tan 3x^\circ} = \lim_{x\to 0}\frac{\sin\left(\frac{\pi x}{180}\right)}{\tan\left(\frac{3\pi x}{180}\right)}

Write each trigonometric function over its argument:

\displaystyle = \lim_{x\to 0}\frac{ \frac{\sin\left(\frac{\pi x}{180}\right)}{\frac{\pi x}{180}} \cdot \frac{\pi x}{180} }{ \frac{\tan\left(\frac{3\pi x}{180}\right)}{\frac{3\pi x}{180}} \cdot \frac{3\pi x}{180} }

x\to 0

\frac{\pi x}{180}\to 0

and

\frac{3\pi x}{180}\to 0

Using the standard limits,

\frac{\sin\left(\frac{\pi x}{180}\right)}{\frac{\pi x}{180}} \to 1

and

\frac{\tan\left(\frac{3\pi x}{180}\right)}{\frac{3\pi x}{180}} \to 1

Thus the limit simplifies to

\displaystyle\lim_{x\to 0}\frac{\frac{\pi x}{180}}{\frac{3\pi x}{180}} = \frac{1}{3}

Answer:

Option B:

\frac{1}{3}

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