Tìm được các giới hạn một bên sau:

a) \(\mathop {\lim }\limits_{x \to {2^ + }} \frac{x}{{x + 1}} = \frac{2}{3}\)

b) \(\mathop {\lim }\limits_{x \to {1^ + }} \frac{{2x - 1}}{{x - 1}} = - \infty \)

c) \(\mathop {\lim }\limits_{x \to {3^ - }} \frac{{{x^2} - 3x}}{{{x^2} - 6x + 9}} = + \infty \)

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

a) \(\mathop {\lim }\limits_{x \to {2^ + }} \frac{x}{{x + 1}} = \frac{2}{{2 + 1}} = \frac{2}{3}\).

b) \(\mathop {\lim }\limits_{x \to {1^ + }} \frac{{2x - 1}}{{x - 1}} = \mathop {\lim }\limits_{x \to {1^ + }} \left[ {(2x - 1) \cdot \frac{1}{{x - 1}}} \right] = + \infty \) (do \(\mathop {\lim }\limits_{x \to {1^ + }} (2x - 1) = 1\) và \(\mathop {\lim }\limits_{x \to {1^ + }} \frac{1}{{x - 1}} = + \infty \)).

c) \(\mathop {\lim }\limits_{x \to {3^ - }} \frac{{{x^2} - 3x}}{{{x^2} - 6x + 9}} = \mathop {\lim }\limits_{x \to {3^ - }} \frac{{x(x - 3)}}{{{{(x - 3)}^2}}} = \mathop {\lim }\limits_{x \to {3^ - }} \frac{x}{{x - 3}} = \mathop {\lim }\limits_{x \to {3^ - }} \left( {x\frac{1}{{x - 3}}} \right) = - \infty \),

do \(\mathop {\lim }\limits_{x \to {3^ - }} x = 3\) và \(\mathop {\lim }\limits_{x \to {3^ - }} \frac{1}{{x - 3}} = - \infty \).

d)

\({\rm{ }}\begin{array}{*{20}{l}}{\mathop {\lim }\limits_{x \to {1^ + }} \left[ {\left( {{x^3} - 1} \right)\left( {\sqrt {\frac{x}{{{x^2} - 1}}} } \right)} \right] = \mathop {\lim }\limits_{x \to {1^ + }} \left[ {(x - 1)\left( {{x^2} + x + 1} \right)\sqrt {\frac{x}{{(x - 1)(x + 1)}}} } \right]}\\{ = \mathop {\lim }\limits_{x \to {1^ + }} \left[ {\left( {{x^2} + x + 1} \right)\sqrt {\frac{{x{{(x - 1)}^2}}}{{(x - 1)(x + 1)}}} } \right] = \mathop {\lim }\limits_{x \to {1^ + }} \left[ {\left( {{x^2} + x + 1} \right)\sqrt {\frac{{x(x - 1)}}{{x + 1}}} } \right] = 3 \cdot \sqrt {\frac{0}{2}} = 0.}\end{array}\)