Câu 42 trang 167 SGK Đại số và Giải tích 11 Nâng cao

Đề bài

Tìm các giới hạn sau :

a.  \(\mathop {\lim }\limits_{x \to 0} \left( {{1 \over x} + {1 \over {{x^2}}}} \right)\)

b.  \(\mathop {\lim }\limits_{x \to - 2} {{{x^3} + 8} \over {x + 2}}\)

c.  \(\mathop {\lim }\limits_{x \to 9} {{3 - \sqrt x } \over {9 - x}}\)

d.  \(\mathop {\lim }\limits_{x \to 0} {{2 - \sqrt {4 - x} } \over x}\)

e.  \(\mathop {\lim }\limits_{x \to + \infty } {{{x^4} - {x^3} + 11} \over {2x - 7}}\)

f.  \(\mathop {\lim }\limits_{x \to - \infty } {{\sqrt {{x^4} + 4} } \over {x + 4}}\)

Hướng dẫn giải

a.  \(\mathop {\lim }\limits_{x \to 0} \left( {{1 \over x} + {1 \over {{x^2}}}} \right) = \mathop {\lim }\limits_{x \to 0} {{x + 1} \over {{x^2}}} = + \infty \)

vì  \(\mathop {\lim }\limits_{x \to 0} \left( {x + 1} \right) = 1 > 0,\mathop {\lim }\limits_{x \to 0} {x^2} = 0\,\text{ và }\,{x^2} > 0,\forall x \ne 0\)

b.

\(\eqalign{
& \mathop {\lim }\limits_{x \to - 2} {{{x^3} + 8} \over {x + 2}} = \mathop {\lim }\limits_{x \to - 2} {{\left( {x + 2} \right)\left( {{x^2} - 2x + 4} \right)} \over {x + 2}} \cr
& = \mathop {\lim }\limits_{x \to - 2} \left( {{x^2} - 2x + 4} \right) = 12 \cr} \)

c.  \(\mathop {\lim }\limits_{x \to 9} {{3 - \sqrt x } \over {9 - x}} = \mathop {\lim }\limits_{x \to 9} {1 \over {3 + \sqrt x }} = {1 \over 6}\)

d.

\(\eqalign{
& \mathop {\lim }\limits_{x \to 0} {{2 - \sqrt {4 - x} } \over x} = \mathop {\lim }\limits_{x \to 0} {{4 - \left( {4 - x} \right)} \over {x\left( {2 + \sqrt {4 - x} } \right)}} \cr
& = \mathop {\lim }\limits_{x \to 0} {1 \over {2 + \sqrt {4 - x} }} = {1 \over 4} \cr} \)

e.  

\(\mathop {\lim }\limits_{x \to + \infty } {{{x^4} - {x^3} + 11} \over {2x - 7}}\)

\(=\mathop {\lim }\limits_{x \to + \infty } {{{x^3} - {x^2} + {{11} \over x}} \over {2 - {7 \over x}}} = + \infty \)

f. Với \(x < 0\), ta có :  \({{\sqrt {{x^4} + 4} } \over {x + 4}} = {{{x^2}\sqrt {1 + {4 \over {{x^4}}}} } \over {x + 4}} = {{x\sqrt {1 + {4 \over {{x^2}}}} } \over {1 + {4 \over x}}}\)

vì  \(\mathop {\lim }\limits_{x \to - \infty } x\sqrt {1 + {4 \over {{x^4}}}} = - \infty \,\text{ và }\,\mathop {\lim }\limits_{x \to - \infty } \left( {1 + {4 \over x}} \right) = 1\)

nên \(\mathop {\lim }\limits_{x \to - \infty } {{\sqrt {{x^4} + 4} } \over {x + 4}} = - \infty \)