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Tìm giới hạn \(I = \mathop {\lim }\limits_{x \to + \infty } \left( {x + 1 - \sqrt {{x^2} - x + 2} } \right)\).

\(I = \frac{1}{2}\).

\(I = \frac{{17}}{{11}}\).

\(I = \frac{{46}}{{31}}\).

\(I = \frac{3}{2}\).

Giải thích

Ta có: \(I = \mathop {\lim }\limits_{x \to + \infty } \left( {x + 1 - \sqrt {{x^2} - x + 2} } \right)\)

\[ = \mathop {\lim }\limits_{x \to + \infty } \left( {\frac{{{x^2} - {x^2} + x - 2}}{{x + \sqrt {{x^2} - x + 2} }} + 1} \right)\] \[ = \mathop {\lim }\limits_{x \to + \infty } \left( {\frac{{x - 2}}{{x + \sqrt {{x^2} - x + 2} }} + 1} \right)\]

\[ = \mathop {\lim }\limits_{x \to + \infty } \left( {\frac{{1 - \frac{2}{x}}}{{1 + \sqrt {1 - \frac{1}{x} + \frac{2}{{{x^2}}}} }} + 1} \right)\]\( = \frac{3}{2}\). Chọn D.