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Bài 24. Tìm các giới hạn sau :
a. \(\mathop {\lim }\limits_{x \to – \infty } {{3{x^2} – x + 7} \over {2{x^3} – 1}}\)
b. \(\mathop {\lim }\limits_{x \to – \infty } {{2{x^4} + 7{x^3} – 15} \over {{x^4} + 1}}\)
c. \(\mathop {\lim }\limits_{x \to + \infty } {{\sqrt {{x^6} + 2} } \over {3{x^3} – 1}}\)
d. \(\mathop {\lim }\limits_{x \to – \infty } {{\sqrt {{x^6} + 2} } \over {3{x^3} – 1}}\)
a.
\(\eqalign{
& \mathop {\lim }\limits_{x \to – \infty } {{3{x^2} – x + 7} \over {2{x^3} – 1}} = \mathop {\lim }\limits_{x \to – \infty } {{{x^3}\left( {{3 \over x} – {1 \over {{x^2}}} + {7 \over {{x^3}}}} \right)} \over {{x^3}\left( {2 – {1 \over {{x^3}}}} \right)}} \cr
& = \mathop {\lim }\limits_{x \to – \infty } {{{3 \over x} – {1 \over {{x^2}}} + {7 \over {{x^3}}}} \over {2 – {1 \over {{x^3}}}}} = {0 \over 2} = 0 \cr} \)
b.
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\(\eqalign{
& \mathop {\lim }\limits_{x \to – \infty } {{2{x^4} + 7{x^3} – 15} \over {{x^4} + 1}} = \mathop {\lim }\limits_{x \to – \infty } {{{x^4}\left( {2 + {7 \over x} – {{15} \over {{x^4}}}} \right)} \over {{x^4}\left( {1 + {1 \over {{x^4}}}} \right)}} \cr
& = \mathop {\lim }\limits_{x \to – \infty } {{2 + {7 \over x} – {{15} \over {{x^4}}}} \over {1 + {1 \over {{x^4}}}}} = 2 \cr} \)
c.
\(\eqalign{
& \mathop {\lim }\limits_{x \to + \infty } {{\sqrt {{x^6} + 2} } \over {3{x^3} – 1}} = \mathop {\lim }\limits_{x \to – \infty } {{{x^3}\sqrt {1 + {2 \over {{x^6}}}} } \over {{x^3}\left( {3 – {1 \over {{x^3}}}} \right)}} \cr
& = \mathop {\lim }\limits_{x \to – \infty } {{\sqrt {1 + {2 \over {{x^6}}}} } \over {3 – {1 \over {{x^3}}}}} = {1 \over 3} \cr} \)
d. Với mọi \(x < 0\), ta có:
\({{\sqrt {{x^6} + 2} } \over {3{x^3} – 1}} = {{\left| x^3 \right|\sqrt {1 + {2 \over {{x^6}}}} } \over {3{x^3} – 1}} = {{ – {x^3}\sqrt {1 + {2 \over {{x^6}}}} } \over {3{x^3} – 1}} = {{ – \sqrt {1 + {2 \over {{x^6}}}} } \over {3 – {1 \over {{x^3}}}}}\)
Do đó :
\(\mathop {\lim }\limits_{x \to – \infty } {{\sqrt {{x^6} + 2} } \over {3{x^3} – 1}} = \mathop {\lim }\limits_{x \to – \infty } {{ – \sqrt {1 + {2 \over {{x^6}}}} } \over {3 – {1 \over {{x^3}}}}} = – {1 \over 3}\)