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Range of Quadratic Expression

Evaluate the ...

Question

Evaluate the limit to infinity
$lim x \to \infty \frac{\sqrt{{3 x}^{2} - 1} + \sqrt{{2 x}^{2} - 1}}{4 x + 3}$

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Solution

$lim x \to \infty \frac{\sqrt{3 x^{2} - 1} + \sqrt{2 x^{2} - 1}}{4 x + 3}$

$= lim x \to \infty \frac{x (\sqrt{3 - \frac{1}{x^{2}}} + \sqrt{2 - \frac{1}{x^{2}}})}{x (4 + \frac{3}{x})}$

$= \frac{\sqrt{3} + \sqrt{2}}{4}$

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