Calculate-1-lim-x-1-cos-2-x-1-x-2-lim-x-e-1-x-1-x-x-x-e-

Question Number 160501 by LEKOUMA last updated on 30/Nov/21

C a l c u l a t e

1) \lim_{x \to 1} \frac{\cos (\frac{Π}{2}) x}{1 - \sqrt{x}}

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

Commented by cortano last updated on 01/Dec/21

(1) \lim_{x \to 1} \frac{\cos (\frac{π}{2} x)}{1 - \sqrt{x}}

let 1 - \sqrt{x} = u; x = {(1 - u)}^{2}

\lim_{u \to 0} \frac{\cos (\frac{π}{2} {(1 - u)}^{2})}{u}

= \lim_{u \to 0} \frac{\sin \frac{π}{2} (u^{2} - 2 u)}{u}

= \lim_{u \to 0} \frac{\frac{π}{2} u (u - 2)}{u} = - π

Answered by qaz last updated on 01/Dec/21

\lim_{x \to 1} \frac{\cos \frac{π}{2} x}{1 - \sqrt{x}} = \lim_{x \to 1} \frac{\cos \frac{π}{2} x}{- \ln \sqrt{x}} = \lim_{x \to 1} \frac{- \frac{π}{2} \sin \frac{π}{2} x}{- \frac{1}{2 x}} = π

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