Tính các giới hạn sau:
a) \(\mathop {\lim }\limits_{x \to - 1} \left( { - 4{x^2} + 3x + 1} \right)\)
b) \(\mathop {\lim }\limits_{x \to - 1} \frac{{ - 4x + 1}}{{{x^2} - x + 3}}\)
c) \(\mathop {\lim }\limits_{x \to 2} \sqrt {3{x^2} + 5x + 4} \)
d) \(\mathop {\lim }\limits_{x \to - \infty } \frac{{ - 3 + \frac{4}{x}}}{{2{x^2} + 3}}\)
e) \(\mathop {\lim }\limits_{x \to {2^ + }} \frac{{ - 3}}{{x - 2}}\)
g) \(\mathop {\lim }\limits_{x \to {{\left( { - 2} \right)}^ + }} \frac{5}{{x + 2}}\)
Sử dụng các định lý về giới hạn hàm số.
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a) Ta có \(\mathop {\lim }\limits_{x \to - 1} \left( { - 4{x^2} + 3x + 1} \right) = \mathop {\lim }\limits_{x \to - 1} \left( { - 4{x^2}} \right) + \mathop {\lim }\limits_{x \to - 1} 3x + \mathop {\lim }\limits_{x \to - 1} 1 = - 4 + \left( { - 3} \right) + 1 = - 6\)
b) Ta có \(\mathop {\lim }\limits_{x \to - 1} \frac{{ - 4x + 1}}{{{x^2} - x + 3}} = \frac{{\mathop {\lim }\limits_{x \to - 1} \left( { - 4x} \right) + \mathop {\lim }\limits_{x \to - 1} 1}}{{\mathop {\lim }\limits_{x \to - 1} {x^2} + \mathop {\lim }\limits_{x \to - 1} \left( { - x} \right) + \mathop {\lim }\limits_{x \to - 1} 3}} = \frac{{4 + 1}}{{1 + 1 + 3}} = 1\)
c) Xét \(\mathop {\lim }\limits_{x \to 2} \left( {3{x^2} + 5x + 4} \right) = \mathop {\lim }\limits_{x \to 2} 3{x^2} + \mathop {\lim }\limits_{x \to 2} 5x + \mathop {\lim }\limits_{x \to 2} 4 = {3.2^2} + 5.2 + 4 = 26\)
Suy ra \(\mathop {\lim }\limits_{x \to 2} \sqrt {3{x^2} + 5x + 4} = \sqrt {26} \).
d) Ta có:\(\mathop {\lim }\limits_{x \to - \infty } \frac{{ - 3 + \frac{4}{x}}}{{2{x^2} + 3}} = \mathop {\lim }\limits_{x \to - \infty } \frac{{{x^2}\left( {\frac{{ - 3}}{{{x^2}}} + \frac{4}{{{x^3}}}} \right)}}{{{x^2}\left( {2 + \frac{3}{{{x^2}}}} \right)}} = \mathop {\lim }\limits_{x \to - \infty } \frac{{\frac{{ - 3}}{{{x^2}}} + \frac{4}{{{x^3}}}}}{{2 + \frac{3}{{{x^2}}}}}\)
\( = \frac{{\mathop {\lim }\limits_{x \to - \infty } \frac{{ - 3}}{{{x^2}}} + \mathop {\lim }\limits_{x \to - \infty } \frac{4}{{{x^3}}}}}{{\mathop {\lim }\limits_{x \to - \infty } 2 + \mathop {\lim }\limits_{x \to - \infty } \frac{3}{{{x^2}}}}} = \frac{{0 + 0}}{{2 + 0}} = 0\)
e) Ta có \(\mathop {\lim }\limits_{x \to {2^ + }} \frac{{ - 3}}{{x - 2}} = - \infty \)
f) Ta có \(\mathop {\lim }\limits_{x \to {{\left( { - 2} \right)}^ + }} \frac{5}{{x + 2}} = - \infty \)