Calculate-the-lim-x-1-x-x-x-x-x-x-

Question Number 141077 by Opredador last updated on 15/May/21

C a l c u l a t e t h e \underset{x \to 1}{l i m} (\frac{\sqrt{x^{x}} - \sqrt{x}}{x^{x} - x}) .

Answered by EDWIN88 last updated on 15/May/21

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

= \lim_{x \to 1} \frac{1}{\sqrt{x^{x}} + \sqrt{x}} = \frac{1}{2}

Commented by EDWIN88 last updated on 15/May/21

Danke nochmal

Commented by Opredador last updated on 15/May/21

Gracias

Answered by mathmax by abdo last updated on 16/May/21

f (x) = \frac{x^{\frac{x}{2}} - x^{\frac{1}{2}}}{x^{x} - x} changement x - 1 = t give

f (x) = f (1 + t) = \frac{{(1 + t)}^{\frac{1 + t}{2}} - {(1 + t)}^{\frac{1}{2}}}{{(1 + t)}^{1 + t} - (1 + t)} (t \to 0) we have

{(1 + t)}^{\frac{1 + t}{2}} \sim 1 + \frac{(1 + t)}{2} t and {(1 + t)}^{\frac{1}{2}} \sim 1 + \frac{t}{2} \Rightarrow

D \sim 1 + \frac{t}{2} + \frac{t^{2}}{2} - 1 - \frac{t}{2} = \frac{t^{2}}{2}

{(1 + t)}^{1 + t} \sim 1 + (1 + t) t \Rightarrow N \sim 1 + (1 + t) t - 1 - t = t^{2} \Rightarrow

f (1 + t) \sim \frac{t^{2}}{{2 t}^{2}} \to \frac{1}{2} \Rightarrow \lim_{x \to 1} f (x) = \frac{1}{2}