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\mathop {\lim }\limits_{x \to \infty } \frac{{\sqrt {{x^2} + 3} - x - 4}}{{\sqrt {{x^2} + 2} + x}} = \mathop {\lim }\limits_{x \to \infty } \frac{{\sqrt {1 + \frac{3}{{{x^2}}}} - \frac{1}{x} - \frac{4}{{{x^2}}}}}{{\sqrt {1 + \frac{2}{{{x^2}}}} + \frac{1}{x}}} = \frac{{\sqrt {1 + 0} - 0 - 0}}{{\sqrt {1 + 0} + 0}} = 1
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