lim-x-3x-2-x-2-x-3-2-

Question Number 83323 by john santu last updated on 01/Mar/20

\lim_{x \to \infty} \sqrt{(3 x - 2) (x - \sqrt{2})} - x \sqrt{3} - \sqrt{2} = ?

Commented by abdomathmax last updated on 01/Mar/20

l e t f (x) = \sqrt{(3 x - 2) (x - \sqrt{2})} - x \sqrt{3} - \sqrt{2}

\Rightarrow f (x) = \sqrt{3 x^{2} - 3 \sqrt{2} x - 2 x + 2 \sqrt{2}} - x \sqrt{3} - \sqrt{2}

= \sqrt{3 x^{2} - (3 \sqrt{2} + 2) x + 2 \sqrt{2}} - x \sqrt{3} - \sqrt{2}

a t + \infty

f (x) = x \sqrt{3} \sqrt{1 - \frac{3 \sqrt{2} + 2}{3 x} + \frac{2 \sqrt{2}}{3 x^{2}}} - x \sqrt{3} - \sqrt{2}

\sim x \sqrt{3} {1 + \frac{1}{2} (- \frac{3 \sqrt{2} + 2}{3 x} + \frac{2 \sqrt{2}}{3 x^{3}})} - x \sqrt{3} - \sqrt{2} \Rightarrow

f (x) \sim \frac{x \sqrt{3}}{2} (- \frac{3 \sqrt{2} + 2}{3 x} + \frac{2 \sqrt{2}}{3 x^{3}}) - \sqrt{2}

\Rightarrow f (x) \sim - \frac{\sqrt{3}}{2} (\frac{3 \sqrt{2} + 2}{3}) + \frac{\sqrt{6}}{3 x^{2}} - \sqrt{2} \Rightarrow

{l i m}_{x \to + \infty} f (x) = - \frac{\sqrt{3}}{6} (3 \sqrt{2} + 2) - \sqrt{2}

a t - \infty {l i m}_{x \to - \infty} f (x) = + \infty

Answered by jagoll last updated on 01/Mar/20

\lim_{x \to \infty} \sqrt{{3 x}^{2} - (3 \sqrt{2} + 2) x + 2 \sqrt{2}} - (\sqrt{{(x \sqrt{3} + \sqrt{2})}^{2}} =

\lim_{x \to \infty} \sqrt{{3 x}^{2} - (3 \sqrt{2} + 2) x + 2 \sqrt{2}} - \sqrt{{3 x}^{2} + 2 \sqrt{6} x + 2} =

\frac{- (3 \sqrt{2} + 2) - 2 \sqrt{6}}{2 \sqrt{3}} = \frac{- 2 \sqrt{6} - 3 \sqrt{2} - 2}{2 \sqrt{3}}

= - \sqrt{2} - \frac{1}{2} \sqrt{6} - \frac{1}{3} \sqrt{3}

Commented by john santu last updated on 01/Mar/20

good sir