Cho hàm số \(f\left( x \right) = \frac{{\sqrt {x + 1} - 2}}{{x - 3}}\).

a) \(f\left( 8 \right) = \frac{{\sqrt {8 + 1} - 2}}{{8 - 3}} = \frac{1}{5}\).

b) \(\mathop {\lim }\limits_{x \to 0} f\left( x \right) = \mathop {\lim }\limits_{x \to 0} \frac{{\sqrt {x + 1} - 2}}{{x - 3}} = \frac{1}{3}\).

c) \(\mathop {\lim }\limits_{x \to 3} f\left( x \right) = \mathop {\lim }\limits_{x \to 3} \frac{{\sqrt {x + 1} - 2}}{{x - 3}} = \mathop {\lim }\limits_{x \to 3} \frac{{x - 3}}{{\left( {x - 3} \right)\left( {\sqrt {x + 1} + 2} \right)}}\)\( = \mathop {\lim }\limits_{x \to 3} \frac{1}{{\sqrt {x + 1} + 2}}\)\( = \frac{1}{4}\).

d) \(\mathop {\lim }\limits_{x \to + \infty } f\left( x \right) = \mathop {\lim }\limits_{x \to + \infty } \frac{{\sqrt {x + 1} - 2}}{{x - 3}}\)\( = \mathop {\lim }\limits_{x \to + \infty } \frac{{\sqrt {\frac{1}{x} + \frac{1}{{{x^2}}}} - \frac{2}{x}}}{{1 - \frac{3}{x}}} = 0\).

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

Suy ra \(a = 0;b = \frac{1}{2}\). Do đó \(3a + 4b = 2\).

Đáp án: a) Sai; b) Đúng; c) Sai; d) Đúng.