lim-x-x-x-2-2x-2-x-2-x-x-

Question Number 84130 by mahdi last updated on 09/Mar/20

\underset{x \to + \infty}{\lim x} (\sqrt{x^{2} + 2 x} - 2 \sqrt{x^{2} + x} + x)

Commented by MJS last updated on 10/Mar/20

let x = \frac{1}{t}

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

= \lim_{t \to 0^{+}} \frac{1 + \sqrt{2 t + 1} - 2 \sqrt{t + 1}}{t^{2}} =

= \lim_{t \to 0^{+}} \frac{\frac{d^{2}}{{d t}^{2}} [1 + \sqrt{2 t + 1} - 2 \sqrt{t + 1}]}{\frac{d^{2}}{{d t}^{2}} [t^{2}]} =

= \lim_{t \to 0^{+}} \frac{1}{2} (\frac{1}{2 {(t + 1)}^{\frac{3}{2}}} - \frac{1}{{(2 t + 1)}^{\frac{3}{2}}}) =

= \frac{1}{2} (\frac{1}{2} - 1) = - \frac{1}{4}

Commented by mahdi last updated on 06/Apr/20

thank you

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