If is equal to then the value of is

JEE MainMathematicsModerateLimits

If
$\ \lim_{x \rightarrow 0}\mspace{2mu}\frac{1 - \cos^{3}x}{xsinx \cdot cosx}\$ is
equal to $\lambda$ then the value of $4\lambda$ is

Show answer and solution

\lim_{x \rightarrow 0}\mspace{2mu}\frac{1 - \cos^{3}x}{xsinx \cdot cosx} = \lim_{x \rightarrow 0}\mspace{2mu}\frac{(1 - cosx) \cdot \left( 1 + \cos^{2}x + cosx \right)}{x^{2} \cdot \left( \frac{sinx}{x} \right) \cdot cosx}

= \lim_{x \rightarrow 0}\mspace{2mu}\left( \frac{1 - \cos x}{x^{2}} \right) \cdot \lim_{x \rightarrow 0}\mspace{2mu}\frac{1}{\left( \frac{\sin x}{x} \right)} \cdot \lim_{x \rightarrow 0}\mspace{2mu}\left( \frac{1 + \cos^{2}x + \cos x}{\cos x} \right)

= \left( \frac{1}{2} \right) \cdot (1) \cdot \left( \frac{3}{1} \right) = \frac{3}{2}

Source: Grass-G / M-19 Mathematics Paper 28 Nov Fourth.docx

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