If the function f ( x ) = sin 3 x + sin x - cos 3 x x 3 , x ? R , is continuous at x = 0 , then f ( 0 ) is equal to [2024]

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Solution

Q.
If the function $f (x) = \frac{\sin 3 x + α \sin x - β \cos 3 x}{x^{3}},$ $x \in R,$ is continuous at $x = 0,$ then $f (0)$ is equal to [2024]

1 $- 4$

2 $- 2$

3 $2$

4 $4$

Ans.
(1)

$f (x) = \frac{\sin 3 x + α \sin x - β \cos 3 x}{x^{3}}$

$f (0) = \lim_{x \to 0} \frac{(3 x - \frac{27 x^{3}}{6} + \dots) + α (x - \frac{x^{3}}{6} + \dots) - β (1 - \frac{9 x^{2}}{2} + \dots)}{x^{3}}$

$= \lim_{x \to 0} \frac{- β + x (3 + α) + x^{2} \frac{9}{2} β + x^{3} (\frac{- 9}{2} - \frac{α}{6})}{x^{3}}$ (Rest of the term will be zero)

Since this limit exists so, $β = 0$

$3 + α = 0$ and $\frac{9}{2} β = 0$

$\Rightarrow α = - 3$

$∴$ $f (0) = \lim_{x \to 0} \frac{(\frac{- 9}{2} - \frac{α}{6}) x^{3}}{x^{3}} = \frac{- 9}{2} + \frac{3}{6}$

$= \frac{- 24}{6} = - 4$

So, $f (0) = - 4$

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