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-

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