Find the value of a for which the limit lim_(x→0) ((sin (ax)−arctan x−x)/(x^3 +x^4 )) is finite and then evaluate the limit


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Question Number 174915 by cortano1 last updated on 14/Aug/22

$F i n d t h e v a l u e o f a f o r w h i c h$ $t h e l i m i t \lim_{x \to 0} \frac{\sin (a x) - \arctan x - x}{x^{3} + x^{4}}$ $i s f i n i t e a n d t h e n e v a l u a t e t h e l i m i t$
	Answered by Ar Brandon last updated on 14/Aug/22

	$\lim_{x \to 0} \frac{\sin (a x) - \arctan x - x}{x^{3} + x^{4}}$ $= \lim_{x \to 0} \frac{(a x - \frac{{(a x)}^{3}}{6}) - (x - \frac{x^{3}}{3}) - x}{x^{3} + x^{4}}$ $[a - 1 - 1 = 0 \Rightarrow a = 2]$ $= \lim_{x \to 0} \frac{- \frac{8}{6} x^{3} + \frac{1}{3} x^{3}}{x^{3} + x^{4}} = \lim_{x \to 0} \frac{- x^{3}}{x^{3} (1 + x)}$ $= \lim_{x \to 0} \frac{- 1}{1 + x} = - 1$
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