Giải thích các bước giải:

\(\begin{array}{l}
a,\\
\mathop {\lim }\limits_{x \to 0} \frac{{{{\left( {x + 3} \right)}^3} - 27}}{x}\\
 = \mathop {\lim }\limits_{x \to 0} \frac{{\left( {{x^3} + 9{x^2} + 27x + 27} \right) - 27}}{x}\\
 = \mathop {\lim }\limits_{x \to 0} \frac{{{x^3} + 9{x^2} + 27x}}{x}\\
 = \mathop {\lim }\limits_{x \to 0} \left( {{x^2} + 9x + 27} \right)\\
 = {0^2} + 9.0 + 27\\
 = 27\\
b,\\
\mathop {\lim }\limits_{x \to 1} \frac{{\sqrt {9 - 5x}  - 2}}{{{x^2} - 1}} = \mathop {\lim }\limits_{x \to 1} \frac{{\left( {\sqrt {9 - 5x}  + 2} \right)\left( {\sqrt {9 - 5x}  - 2} \right)}}{{\left( {\sqrt {9 - 5x}  + 2} \right)\left( {{x^2} - 1} \right)}}\\
 = \mathop {\lim }\limits_{x \to 1} \frac{{\left( {9 - 5x} \right) - {2^2}}}{{\left( {\sqrt {9 - 5x}  + 2} \right)\left( {x - 1} \right)\left( {x + 1} \right)}}\\
 = \mathop {\lim }\limits_{x \to 1} \frac{{5\left( {1 - x} \right)}}{{\left( {\sqrt {9 - 5x}  + 2} \right)\left( {x - 1} \right)\left( {x + 1} \right)}}\\
 = \mathop {\lim }\limits_{x \to 1} \frac{{ - 5}}{{\left( {\sqrt {9 - 5x}  + 2} \right)\left( {x + 1} \right)}}\\
 = \frac{{ - 5}}{{\left( {\sqrt {9 - 5.1}  + 2} \right)\left( {1 + 1} \right)}}\\
 = \frac{{ - 5}}{8}\\
c,\\
\mathop {\lim }\limits_{x \to 1} \frac{{\sqrt {2x + 7}  - 3}}{{2 - \sqrt {x + 3} }}\\
 = \mathop {\lim }\limits_{x \to 1} \frac{{\left( {\sqrt {2x + 7}  - 3} \right)\left( {\sqrt {2x + 7}  + 3} \right)\left( {2 + \sqrt {x + 3} } \right)}}{{\left( {2 - \sqrt {x + 3} } \right)\left( {2 + \sqrt {x + 3} } \right)\left( {\sqrt {2x + 7}  + 3} \right)}}\\
 = \mathop {\lim }\limits_{x \to 1} \frac{{\left[ {\left( {2x + 7} \right) - {3^2}} \right]\left( {2 + \sqrt {x + 3} } \right)}}{{\left[ {{2^2} - \left( {x + 3} \right)} \right]\left( {\sqrt {2x + 7}  + 3} \right)}}\\
 = \mathop {\lim }\limits_{x \to 1} \frac{{2\left( {x - 1} \right)\left( {2 + \sqrt {x + 3} } \right)}}{{\left( {1 - x} \right)\left( {\sqrt {2x + 7}  + 3} \right)}}\\
 = \mathop {\lim }\limits_{x \to 1} \frac{{2.\left( {2 + \sqrt {x + 3} } \right)}}{{ - \left( {\sqrt {2x + 7}  + 3} \right)}}\\
 = \frac{{2.\left( {2 + \sqrt {1 + 3} } \right)}}{{ - \left( {\sqrt {2.1 + 7}  + 3} \right)}}\\
 = \frac{{2.4}}{{ - 6}} =  - \frac{4}{3}\\
d,\\
\mathop {\lim }\limits_{x \to 2} \frac{{\sqrt[3]{{3 - x}} - 1}}{{x - 2}} = \mathop {\lim }\limits_{x \to 2} \frac{{\left( {\sqrt[3]{{3 - x}} - 1} \right)\left( {{{\sqrt[3]{{3 - x}}}^2} + \sqrt[3]{{3 - x}}.1 + {1^2}} \right)}}{{\left( {x - 2} \right).\left( {{{\sqrt[3]{{3 - x}}}^2} + \sqrt[3]{{3 - x}}.1 + {1^2}} \right)}}\\
 = \mathop {\lim }\limits_{x \to 2} \frac{{\left( {3 - x} \right) - {1^3}}}{{\left( {x - 2} \right)\left( {{{\sqrt[3]{{3 - x}}}^2} + \sqrt[3]{{3 - x}} + 1} \right)}} = \mathop {\lim }\limits_{x \to 2} \frac{{2 - x}}{{\left( {x - 2} \right)\left( {{{\sqrt[3]{{3 - x}}}^2} + \sqrt[3]{{3 - x}} + 1} \right)}}\\
 = \mathop {\lim }\limits_{x \to 2} \frac{{ - 1}}{{{{\sqrt[3]{{3 - x}}}^2} + \sqrt[3]{{3 - x}} + 1}} = \frac{{ - 1}}{{{{\sqrt[3]{{3 - 2}}}^2} + \sqrt[3]{{3 - 2}} + 1}} = \frac{{ - 1}}{3}\\
e,\\
\mathop {\lim }\limits_{x \to 1} \frac{{\sqrt {5 - x}  - \sqrt[3]{{{x^2} + 7}}}}{{{x^2} - 1}}\\
 = \mathop {\lim }\limits_{x \to 1} \frac{{\left( {\sqrt {5 - x}  - 2} \right) + \left( {2 - \sqrt[3]{{{x^2} + 7}}} \right)}}{{\left( {x - 1} \right)\left( {x + 1} \right)}}\\
 = \mathop {\lim }\limits_{x \to 1} \frac{{\frac{{\left( {5 - x} \right) - {2^2}}}{{\sqrt {5 - x}  + 2}} + \frac{{{2^3} - \left( {{x^2} + 7} \right)}}{{{2^2} + 2.\sqrt[3]{{{x^2} + 7}} + {{\sqrt[3]{{{x^2} + 7}}}^2}}}}}{{\left( {x - 1} \right)\left( {x + 1} \right)}}\\
 = \mathop {\lim }\limits_{x \to 1} \frac{{\frac{{1 - x}}{{\sqrt {5 - x}  + 2}} + \frac{{1 - {x^2}}}{{4 + 2.\sqrt[3]{{{x^2} + 7}} + {{\sqrt[3]{{{x^2} + 7}}}^2}}}}}{{\left( {x - 1} \right)\left( {x + 1} \right)}}\\
 = \mathop {\lim }\limits_{x \to 1} \left[ {\frac{{ - 1}}{{\left( {x + 1} \right).\left( {\sqrt {5 - x}  + 2} \right)}} + \frac{{ - 1}}{{4 + 2.\sqrt[3]{{{x^2} + 7}} + {{\sqrt[3]{{{x^2} + 7}}}^2}}}} \right]\\
 = \frac{{ - 1}}{{\left( {1 + 1} \right).\left( {\sqrt {5 - 1}  + 2} \right)}} + \frac{{ - 1}}{{4 + 2.\sqrt[3]{{{1^2} + 7}} + {{\sqrt[3]{{{1^2} + 7}}}^2}}}\\
 =  - \frac{5}{{24}}
\end{array}\)