If-lim-x-0-sin-2x-asin-x-x-3-exist-what-is-the-value-of-a-and-the-limit-

Question Number 112497 by bemath last updated on 08/Sep/20

If \lim_{x \to 0} \frac{\sin 2 x + asin x}{x^{3}} exist

what is the value of a and the limit ?

Answered by mathmax by abdo last updated on 08/Sep/20

let f (x) = \frac{\sin (2 x) + asinx}{x^{3}} we have sinx \sim x - \frac{x^{3}}{6}

\sin (2 x) \sim 2 x - \frac{{(2 x)}^{3}}{6} = 2 x - \frac{4}{3} x^{3} \Rightarrow f (x) \sim \frac{2 x - \frac{4}{3} x^{3} + ax - \frac{{ax}^{3}}{6}}{x^{3}}

= \frac{2 + a - x^{2} (\frac{4}{3} + \frac{a}{6})}{x^{2}} so \lim_{x \to 0} f (x) exist \Leftrightarrow 2 + a = 0 \Leftrightarrow a = - 2

and the limit is L = - (\frac{4}{3} - \frac{1}{3}) = - 1

Answered by bemath last updated on 08/Sep/20

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