Giải thích các bước giải:

\(\begin{array}{l}
1,\\
\mathop {\lim }\limits_{x \to 1} \frac{{\sqrt {x + 3}  + \sqrt {3x + 1}  - 4}}{{1 - {x^2}}}\\
 = \mathop {\lim }\limits_{x \to 1} \frac{{\left( {\sqrt {x + 3}  - 2} \right) + \left( {\sqrt {3x + 1}  - 2} \right)}}{{1 - {x^2}}}\\
 = \mathop {\lim }\limits_{x \to 1} \dfrac{{\frac{{\left( {x + 3} \right) - {2^2}}}{{\sqrt {x + 3}  + 2}} + \frac{{\left( {3x + 1} \right) - {2^2}}}{{\sqrt {3x + 1}  + 2}}}}{{\left( {1 - x} \right)\left( {1 + x} \right)}}\\
 = \mathop {\lim }\limits_{x \to 1} \dfrac{{\frac{{x - 1}}{{\sqrt {x + 3}  + 2}} + \frac{{3x - 3}}{{\sqrt {3x + 1}  + 2}}}}{{\left( {1 - x} \right)\left( {1 + x} \right)}}\\
 = \mathop {\lim }\limits_{x \to 1} \dfrac{{\frac{1}{{\sqrt {x + 3}  + 2}} + \frac{3}{{\sqrt {3x + 1}  + 2}}}}{{ - \left( {x + 1} \right)}}\\
 = \dfrac{{\frac{1}{{\sqrt {1 + 3}  + 2}} + \frac{3}{{\sqrt {3.1 + 1}  + 2}}}}{{ - \left( {1 + 1} \right)}}\\
 =  - \frac{1}{2}\\
b,\\
\mathop {\lim }\limits_{x \to 7} \frac{{\sqrt {x + 2}  - \sqrt[3]{{4x - 1}}}}{{{x^2} - 49}}\\
 = \mathop {\lim }\limits_{x \to 7} \frac{{\left( {\sqrt {x + 2}  - 3} \right) + \left( {3 - \sqrt[3]{{4x - 1}}} \right)}}{{\left( {x - 7} \right)\left( {x + 7} \right)}}\\
 = \mathop {\lim }\limits_{x \to 7} \dfrac{{\frac{{\left( {x + 2} \right) - {3^2}}}{{\sqrt {x + 2}  + 3}} + \frac{{{3^3} - \left( {4x - 1} \right)}}{{{3^2} + 3.\sqrt[3]{{4x - 1}} + {{\sqrt[3]{{4x - 1}}}^2}}}}}{{\left( {x - 7} \right)\left( {x + 7} \right)}}\\
 = \mathop {\lim }\limits_{x \to 7} \dfrac{{\frac{{x - 7}}{{\sqrt {x + 2}  + 3}} + \frac{{28 - 4x}}{{{{\sqrt[3]{{4x - 1}}}^2} + 3.\sqrt[3]{{4x - 1}} + 9}}}}{{\left( {x - 7} \right)\left( {x + 7} \right)}}\\
 = \mathop {\lim }\limits_{x \to 7} \dfrac{{\frac{1}{{\sqrt {x + 2}  + 3}} - \frac{4}{{{{\sqrt[3]{{4x - 1}}}^2} + 3.\sqrt[3]{{4x - 1}} + 9}}}}{{x + 7}}\\
 = \dfrac{{\frac{1}{{\sqrt {7 + 2}  + 3}} - \frac{4}{{{{\sqrt[3]{{4.7 - 1}}}^2} + 3.\sqrt[3]{{4.7 - 1}} + 9}}}}{{7 + 7}}\\
 = \frac{1}{{756}}\\
c,\\
\mathop {\lim }\limits_{x \to 0} \frac{{\sqrt {4x + 1}  - \sqrt[3]{{6x + 1}}}}{{{x^2}}}\\
 = \mathop {\lim }\limits_{x \to 0} \frac{{\left( {\sqrt {4x + 1}  - \left( {2x + 1} \right)} \right) + \left( {\left( {2x + 1} \right) - \sqrt[3]{{6x + 1}}} \right)}}{{{x^2}}}\\
 = \mathop {\lim }\limits_{x \to 0} \dfrac{{\frac{{\left( {4x + 1} \right) - {{\left( {2x + 1} \right)}^2}}}{{\sqrt {4x + 1}  + 2x + 1}} + \frac{{{{\left( {2x + 1} \right)}^3} - \left( {6x + 1} \right)}}{{{{\left( {2x + 1} \right)}^2} + \left( {2x + 1} \right).\sqrt[3]{{6x + 1}} + {{\sqrt[3]{{6x + 1}}}^2}}}}}{{{x^2}}}\\
 = \mathop {\lim }\limits_{x \to 0} \dfrac{{\frac{{4x + 1 - \left( {4{x^2} + 4x + 1} \right)}}{{\sqrt {4x + 1}  + 2x + 1}} + \frac{{8{x^3} + 12{x^2} + 6x + 1 - \left( {6x + 1} \right)}}{{{{\left( {2x + 1} \right)}^2} + \left( {2x + 1} \right).\sqrt[3]{{6x + 1}} + {{\sqrt[3]{{6x + 1}}}^2}}}}}{{{x^2}}}\\
 = \mathop {\lim }\limits_{x \to 0} \frac{{\frac{{ - 4{x^2}}}{{\sqrt {4x + 1}  + 2x + 1}} + \frac{{8{x^3} + 12{x^2}}}{{{{\left( {2x + 1} \right)}^2} + \left( {2x + 1} \right).\sqrt[3]{{6x + 1}} + {{\sqrt[3]{{6x + 1}}}^2}}}}}{{{x^2}}}\\
 = \mathop {\lim }\limits_{x \to 0} \left[ {\frac{{ - 4}}{{\sqrt {4x + 1}  + 2x + 1}} + \frac{{8x + 12}}{{{{\left( {2x + 1} \right)}^2} + \left( {2x + 1} \right).\sqrt[3]{{6x + 1}} + {{\sqrt[3]{{6x + 1}}}^2}}}} \right]\\
 = \frac{{ - 4}}{{\sqrt {4.0 + 1}  + 2.0 + 1}} + \frac{{8.0 + 12}}{{{{\left( {2.0 + 1} \right)}^2} + \left( {2.0 + 1} \right).\sqrt[3]{{6.0 + 1}} + {{\sqrt[3]{{6.0 + 1}}}^2}}}\\
 = \frac{{ - 4}}{2} + \frac{{12}}{3} = 2
\end{array}\)

\(\begin{array}{l}
d,\\
\mathop {\lim }\limits_{x \to  - 1} \frac{{\sqrt {5 + 4x}  - \sqrt[3]{{7 + 6x}}}}{{{x^3} + {x^2} - x - 1}}\\
 = \mathop {\lim }\limits_{x \to  - 1} \frac{{\left( {\sqrt {5 + 4x}  - \left( {2x + 3} \right)} \right) + \left( {\left( {2x + 3} \right) - \sqrt[3]{{7 + 6x}}} \right)}}{{{x^2}\left( {x + 1} \right) - \left( {x + 1} \right)}}\\
 = \mathop {\lim }\limits_{x \to  - 1} \dfrac{{\frac{{\left( {5 + 4x} \right) - {{\left( {2x + 3} \right)}^2}}}{{\sqrt {5 + 4x}  + \left( {2x + 3} \right)}} + \frac{{{{\left( {2x + 3} \right)}^3} - \left( {7 + 6x} \right)}}{{{{\left( {2x + 3} \right)}^2} + \left( {2x + 3} \right).\sqrt[3]{{7 + 6x}} + {{\sqrt[3]{{7 + 6x}}}^2}}}}}{{\left( {x + 1} \right)\left( {{x^2} - 1} \right)}}\\
 = \mathop {\lim }\limits_{x \to  - 1} \dfrac{{\frac{{\left( {5 + 4x} \right) - \left( {4{x^2} + 12x + 9} \right)}}{{\sqrt {5 + 4x}  + \left( {2x + 3} \right)}} + \frac{{\left( {8{x^3} + 36{x^2} + 54x + 27} \right) - \left( {7 + 6x} \right)}}{{{{\left( {2x + 3} \right)}^2} + \left( {2x + 3} \right).\sqrt[3]{{7 + 6x}} + {{\sqrt[3]{{7 + 6x}}}^2}}}}}{{\left( {x - 1} \right){{\left( {x + 1} \right)}^2}}}\\
 = \mathop {\lim }\limits_{x \to  - 1} \dfrac{{\frac{{ - 4{x^2} - 8x - 4}}{{\sqrt {5 + 4x}  + \left( {2x + 3} \right)}} + \frac{{8{x^3} + 36{x^2} + 48x + 20}}{{{{\left( {2x + 3} \right)}^2} + \left( {2x + 3} \right).\sqrt[3]{{7 + 6x}} + {{\sqrt[3]{{7 + 6x}}}^2}}}}}{{\left( {x - 1} \right){{\left( {x + 1} \right)}^2}}}\\
 = \mathop {\lim }\limits_{x \to  - 1} \dfrac{{\frac{{ - 4{{\left( {x + 1} \right)}^2}}}{{\sqrt {5 + 4x}  + \left( {2x + 3} \right)}} + \frac{{4\left( {2x + 5} \right){{\left( {x + 1} \right)}^2}}}{{{{\left( {2x + 3} \right)}^2} + \left( {2x + 3} \right).\sqrt[3]{{7 + 6x}} + {{\sqrt[3]{{7 + 6x}}}^2}}}}}{{\left( {x - 1} \right){{\left( {x + 1} \right)}^2}}}\\
 = \mathop {\lim }\limits_{x \to  - 1} \dfrac{{\frac{{ - 4}}{{\sqrt {5 + 4x}  + \left( {2x + 3} \right)}} + \frac{{4\left( {2x + 5} \right)}}{{{{\left( {2x + 3} \right)}^2} + \left( {2x + 3} \right).\sqrt[3]{{7 + 6x}} + {{\sqrt[3]{{7 + 6x}}}^2}}}}}{{x - 1}}\\
 = \dfrac{{\frac{{ - 4}}{{\sqrt {5 - 4.1}  + 2.\left( { - 1} \right) + 3}} + \frac{{4.\left( {2.\left( { - 1} \right) + 5} \right)}}{{{{\left( {2.\left( { - 1} \right) + 3} \right)}^2} + \left( {2.\left( { - 1} \right) + 3} \right).\sqrt[3]{{7 - 6.1}} + {{\sqrt[3]{{7 - 6.1}}}^2}}}}}{{ - 1 - 1}}\\
 =  - 1\\
e,\\
\mathop {\lim }\limits_{x \to 2} \frac{{\sqrt {x + 2} .\sqrt[3]{{x - 1}} - 2}}{{\sqrt {5 - 2x}  - 1}}\\
 = \mathop {\lim }\limits_{x \to 2} \frac{{\sqrt {x + 2} .\left( {\sqrt[3]{{x - 1}} - 1} \right) + \left( {\sqrt {x + 2}  - 2} \right)}}{{\sqrt {5 - 2x}  - 1}}\\
 = \mathop {\lim }\limits_{x \to 2} \dfrac{{\sqrt {x + 2} .\frac{{\left( {x - 1} \right) - 1}}{{{{\sqrt[3]{{x - 1}}}^2} + \sqrt[3]{{x - 1}}.1 + {1^2}}} + \frac{{\left( {x + 2} \right) - {2^2}}}{{\sqrt {x + 2}  + 2}}}}{{\frac{{\left( {5 - 2x} \right) - {1^2}}}{{\sqrt {5 - 2x}  + 1}}}}\\
 = \mathop {\lim }\limits_{x \to 2} \dfrac{{\sqrt {x + 2} .\frac{{x - 2}}{{{{\sqrt[3]{{x - 1}}}^2} + \sqrt[3]{{x - 1}} + 1}} + \frac{{x - 2}}{{\sqrt {x + 2}  + 2}}}}{{\frac{{2\left( {2 - x} \right)}}{{\sqrt {5 - 2x}  + 1}}}}\\
 = \mathop {\lim }\limits_{x \to 2} \dfrac{{\sqrt {x + 2} .\frac{1}{{{{\sqrt[3]{{x - 1}}}^2} + \sqrt[3]{{x - 1}} + 1}} + \frac{1}{{\sqrt {x + 2}  + 2}}}}{{\frac{{ - 2}}{{\sqrt {5 - 2x}  + 1}}}}\\
 = \dfrac{{\frac{{\sqrt {2 + 2} }}{{{{\sqrt[3]{{2 - 1}}}^2} + \sqrt[3]{{2 - 1}} + 1}} + \frac{1}{{\sqrt {2 + 2}  + 2}}}}{{\frac{{ - 2}}{{\sqrt {5 - 2.2}  + 1}}}}\\
 =  - \frac{{11}}{{12}}
\end{array}\)