lim-x-x-x-2-x-1-ln-e-x-x-x-

Question Number 79565 by jagoll last updated on 26/Jan/20

\lim_{x \to \infty} (x - \sqrt{x^{2} - x + 1}) \times (\frac{\ln (e^{x} + x)}{x})

Commented by john santu last updated on 26/Jan/20

(1) \lim_{x \to \infty} \sqrt{x^{2}} - \sqrt{x^{2} - x + 1} = \frac{0 - (- 1)}{2} = \frac{1}{2}

(2) \lim_{x \to \infty} \frac{\ln (e^{x} + x)}{x} = \lim_{x \to \infty} \frac{e^{x} + 1}{e^{x} + x} =

\lim_{x \to \infty} \frac{e^{x}}{e^{x}} = 1

∴ \frac{1}{2} \times 1 = \frac{1}{2}

Commented by mathmax by abdo last updated on 26/Jan/20

l e t f (x) = (x - \sqrt{x^{2} - x + 1}) (\frac{l n (e^{x} + x)}{x}) \Rightarrow f o r x > 0

f (x) = (x - \sqrt{x^{2} (1 - \frac{1}{x} + \frac{1}{x^{2}}}) (\frac{l n (e^{x} (1 + {x e}^{- x})}{x})

= (x - x \sqrt{1 + \frac{1}{x^{2}} - \frac{1}{x}}) (1 + \frac{l n (1 + {x e}^{- x})}{x})

\sim (x - x (1 + \frac{1}{2} (\frac{1}{x^{2}} - \frac{1}{x})) (1 + \frac{{x e}^{- x}}{x})

= (- \frac{1}{2 x} + \frac{1}{2}) (1 + e^{- x}) \Rightarrow {l i m}_{x \to + \infty} f (x) = \frac{1}{2}