solve-L-lim-x-0-e-x-1-1-x-x-2-

Question Number 70364 by 20190927 last updated on 03/Oct/19

solve L = \lim_{x \to 0} \frac{e^{x} - \frac{1}{1 - x}}{x^{2}}

Commented by kaivan.ahmadi last updated on 03/Oct/19

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

{l i m}_{x \to 0} \frac{- e^{x} - {x e}^{x}}{2 - 6 x} = \frac{- 1}{2}

Commented by 20190927 last updated on 03/Oct/19

Thank you

Commented by mathmax by abdo last updated on 03/Oct/19

l e t f (x) = \frac{e^{x} - \frac{1}{1 - x}}{x^{2}} \Rightarrow f (x) = \frac{(1 - x) e^{x} - 1}{x^{2} (1 - x)}

w e h a v e e^{x} = 1 + x + \frac{x^{2}}{2} + o (x^{3}) \Rightarrow (1 - x) e^{x} = (1 - x) (1 + x + \frac{x^{2}}{2} + o (x^{3}))

= 1 + x + \frac{x^{2}}{2} - x - x^{2} - \frac{x^{3}}{2} + o (x^{4}) = 1 - \frac{x^{2}}{2} - \frac{x^{3}}{2} + o (x^{4}) \Rightarrow

f (x) \sim \frac{- \frac{x^{2}}{2} - \frac{x^{3}}{2}}{x^{2} (1 - x)} \Rightarrow f (x) \sim \frac{- \frac{1}{2} - \frac{x}{2}}{1 - x} (x \to 0) \Rightarrow

{l i m}_{x \to 0} f (x) = - \frac{1}{2}