fimd-lim-x-0-sinx-x-1-x-1-x-x-2-

Question Number 29458 by prof Abdo imad last updated on 08/Feb/18

f i m d {l i m}_{x \to 0} \frac{\frac{s i n x}{x (1 + x)} - 1 + x}{x^{2}} .

Commented by prof Abdo imad last updated on 13/Feb/18

= {l i m}_{x \to 0} \frac{s i n x + x (x - 1) (x + 1)}{x^{3} (1 + x)}

= {l i m}_{x \to 0} \frac{s i n x + x (x^{2} - 1)}{x^{3} (1 + x)}

= {l i m}_{x \to 0} \frac{s i n x + x^{3} - x}{x^{3} (1 + x)} b u t f o r x \in V (0)

s i n x \sim x \Rightarrow \frac{s i n x + x^{3} - x}{x^{3} (1 + x)} \sim \frac{1}{1 + x}

\Rightarrow l i m (\dots) = 1 .