lim-x-e-1-x-cos-1-x-1-1-1-x-2-lim-x-a-x-x-a-a-x-a-

Question Number 165392 by LEKOUMA last updated on 31/Jan/22

\lim_{x \to + \infty} \frac{e^{\frac{1}{x}} - \cos \frac{1}{x}}{1 - \sqrt{1 - \frac{1}{x^{2}}}}

\lim_{x \to a} \frac{x^{x} - a^{a}}{x - a}

Answered by Mathspace last updated on 31/Jan/22

f (x) = \frac{e^{\frac{1}{x}} - c o s (\frac{1}{x})}{1 - \sqrt{1 - \frac{1}{x^{2}}}}

c h a m n g e m e n t \frac{1}{x} = t g i v e

f (x) = f (\frac{1}{t}) = \frac{e^{t} - c o s t}{1 + \sqrt{1 - t^{2}}}

\sim \frac{1 + t - (1 - \frac{t^{2}}{2})}{1 - (1 - \frac{t^{2}}{2})} = \frac{t + \frac{t^{2}}{2}}{\frac{t^{2}}{2}} = \frac{2}{t} + 1 \to \infty (t \to 0)

\Rightarrow {l i m}_{x \to + \infty} f (x) = \infty

Answered by Mathspace last updated on 31/Jan/22

2)

l e t u s e h o s p i t a l

{l i m}_{x \to a} \frac{x^{x} - a^{a}}{x - a}

= {l i m}_{x \to a} {(e^{x l n x})}^{(1)}

= {l i m}_{x \to a} ((l n x + 1) e^{x l n x}

= (1 + l n a) a^{a}