Câu 38. Tìm các giới hạn sau :
a. \(\mathop {\lim }\limits_{x \to 2} {{{x^3} – 8} \over {{x^2} – 4}}\)
b. \(\mathop {\lim }\limits_{x \to {{\left( { – 3} \right)}^ + }} {{2{x^2} + 5x – 3} \over {{{\left( {x + 3} \right)}^2}}}\)
c. \(\mathop {\lim }\limits_{x \to {{\left( { – 3} \right)}^ – }} {{2{x^2} + 5x – 3} \over {{{\left( {x + 3} \right)}^2}}}\)
d. \(\mathop {\lim }\limits_{x \to 0} {{\sqrt {{x^3} + 1} – 1} \over {{x^2} + x}}\)
a. Dạng \({0 \over 0}\) ta phân tích tử và mẫu ra thừa số :
\(\eqalign{
& \mathop {\lim }\limits_{x \to 2} {{{x^3} – 8} \over {{x^2} – 4}} = \mathop {\lim }\limits_{x \to 2} {{\left( {x – 2} \right)\left( {{x^2} + 2x + 4} \right)} \over {\left( {x – 2} \right)\left( {x + 2} \right)}} \cr
& = \mathop {\lim }\limits_{x \to 2} {{{x^2} + 2x + 4} \over {x + 2}} = 3 \cr} \)
b.
\(\eqalign{
& \mathop {\lim }\limits_{x \to {{\left( { – 3} \right)}^ + }} {{2{x^2} + 5x – 3} \over {{{\left( {x + 3} \right)}^2}}} = \mathop {\lim }\limits_{x \to {{\left( { – 3} \right)}^ + }} {{\left( {x + 3} \right)\left( {2x – 1} \right)} \over {{{\left( {x + 3} \right)}^2}}} \cr
& = \mathop {\lim }\limits_{x \to {{\left( { – 3} \right)}^ + }} {{2x – 1} \over {x + 3}} = – \infty \cr} \)
Vì \(\mathop {\lim }\limits_{x \to {{\left( { – 3} \right)}^ + }} \left( {2x – 1} \right) = – 7 < 0\,\text{ và }\,\mathop {\lim }\limits_{x \to {{\left( { – 3} \right)}^ +}} \left( {x + 3} \right) = 0;\)
\(x + 3 > 0\)
c.
\(\eqalign{
& \mathop {\lim }\limits_{x \to {{\left( { – 3} \right)}^ – }} {{2{x^2} + 5x – 3} \over {{{\left( {x + 3} \right)}^2}}} = \mathop {\lim }\limits_{x \to {{\left( { – 3} \right)}^ – }} {{\left( {x + 3} \right)\left( {2x – 1} \right)} \over {{{\left( {x + 3} \right)}^2}}} \cr
& = \mathop {\lim }\limits_{x \to {{\left( { – 3} \right)}^ – }} {{2x – 1} \over {x + 3}} = + \infty \cr} \)
Vì \(\mathop {\lim }\limits_{x \to {{\left( { – 3} \right)}^ – }} \left( {2x – 1} \right) = – 7 < 0\,\text{ và }\,\mathop {\lim }\limits_{x \to {{\left( { – 3} \right)}^ – }} \left( {x + 3} \right) = 0;\)
\(x + 3 < 0\)
d.
\(\eqalign{
& \mathop {\lim }\limits_{x \to 0} {{\sqrt {{x^3} + 1} – 1} \over {{x^2} + x}} = \mathop {\lim }\limits_{x \to 0} {{{x^3}} \over {x\left( {x + 1} \right)\left( {\sqrt {{x^3} + 1} + 1} \right)}} \cr
& = \mathop {\lim }\limits_{x \to 0} {{{x^2}} \over {\left( {x + 1} \right)\left( {\sqrt {{x^3} + 1} + 1} \right)}} = 0 \cr} \)
Câu 39. Tìm các giới hạn sau :
a. \(\mathop {\lim }\limits_{x \to + \infty } {{2{x^2} + x – 10} \over {9 – 3{x^3}}}\)
b. \(\mathop {\lim }\limits_{x \to – \infty } {{\sqrt {2{x^2} – 7x + 12} } \over {3\left| x \right| – 17}}\)
a. \(\mathop {\lim }\limits_{x \to + \infty } {{2{x^2} + x – 10} \over {9 – 3{x^3}}} = \mathop {\lim }\limits_{x \to + \infty } {{{2 \over x} + {1 \over {{x^2}}} – {{10} \over {{x^3}}}} \over {{9 \over {{x^3}}} – 3}} = 0\)
b. Với mọi \(x ≠ 0\), ta có :
\({{\sqrt {2{x^2} – 7x + 12} } \over {3\left| x \right| – 17}} = {{\left| x \right|\sqrt {2 – {7 \over x} + {{12} \over {{x^2}}}} } \over {\left| x \right|\left( {3 – {{17} \over {\left| x \right|}}} \right)}} = {{\sqrt {2 – {7 \over x} + {{12} \over {{x^2}}}} } \over {3 – {{17} \over {\left| x \right|}}}}\)
Do đó \(\mathop {\lim }\limits_{x \to – \infty } {{\sqrt {2{x^2} – 7x + 12} } \over {3\left| x \right| – 17}} = {{\sqrt 2 } \over 3}\)
Câu 40. Tìm các giới hạn sau :
a. \(\mathop {\lim }\limits_{x \to {{\left( { – 1} \right)}^ + }} \left( {{x^3} + 1} \right)\sqrt {{x \over {{x^2} – 1}}} \)
b. \(\mathop {\lim }\limits_{x \to + \infty } \left( {x + 2} \right)\sqrt {{{x – 1} \over {{x^3} + x}}} \)
a. Dạng 0.∞
Với \(x > -1\) đủ gần -1 (\(-1 < x < 0\)) ta có :
\(\eqalign{
& \left( {{x^3} + 1} \right)\sqrt {{x \over {{x^2} – 1}}} \cr &= \left( {{x^2} – x + 1} \right)\left( {x + 1} \right).\sqrt {{x \over {{x^2} – 1}}} \cr
& = \left( {{x^2} – x + 1} \right)\sqrt {{{x\left( {x + 1} \right)} \over {x – 1}}} \cr
& \Rightarrow \mathop {\lim }\limits_{x \to {{\left( { – 1} \right)}^ + }} \left( {{x^3} + 1} \right)\sqrt {{x \over {{x^2} – 1}}}\cr & \;\;= \mathop {\lim }\limits_{x \to {{\left( { – 1} \right)}^ + }} \left( {{x^2} – x + 1} \right)\sqrt {{{x\left( {x + 1} \right)} \over {x – 1}}} = 0 \cr} \)
b. Dạng 0.∞
\(\eqalign{
& \mathop {\lim }\limits_{x \to + \infty } \left( {x + 2} \right)\sqrt {{{x – 1} \over {{x^3} + x}}} \cr &= \mathop {\lim }\limits_{x \to + \infty } \sqrt {{{{{\left( {x + 2} \right)}^2}\left( {x – 1} \right)} \over {{x^3} + x}}} \cr
& = \mathop {\lim }\limits_{x \to + \infty } \sqrt {{{{{\left( {1 + {2 \over x}} \right)}^2}\left( {1 – {1 \over x}} \right)} \over {1 + {1 \over {{x^2}}}}}} = 1 \cr} \)
Câu 41. Tìm các giới hạn sau :
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a. \(\mathop {\lim }\limits_{x \to + \infty } \left( {\sqrt {{x^2} + 1} – x} \right)\)
b. \(\mathop {\lim }\limits_{x \to 1} {{\sqrt {2x – {x^2}} – 1} \over {{x^2} – x}}\)
a. Dạng ∞ – ∞
\(\eqalign{
& \mathop {\lim }\limits_{x \to + \infty } \left( {\sqrt {{x^2} + 1} – x} \right) = \mathop {\lim }\limits_{x \to + \infty } {{{x^2} + 1 – {x^2}} \over {\sqrt {{x^2} + 1} + x}} \cr
& = \mathop {\lim }\limits_{x \to + \infty } {1 \over {\sqrt {{x^2} + 1} + x}} = 0 \cr} \)
b. Dạng \({0 \over 0}\)
\(\eqalign{
& \mathop {\lim }\limits_{x \to 1} {{\sqrt {2x – {x^2}} – 1} \over {{x^2} – x}} = \mathop {\lim }\limits_{x \to 1} {{2x – {x^2} – 1} \over {x\left( {x – 1} \right)\left( {\sqrt {2x – {x^2}} + 1} \right)}} \cr
& = \mathop {\lim }\limits_{x \to 1} {{ – {{\left( {x – 1} \right)}^2}} \over {x\left( {x – 1} \right)\left( {\sqrt {2x – {x^2}} + 1} \right)}} = \mathop {\lim }\limits_{x \to 1} {{1 – x} \over {x\left( {\sqrt {2x – {x^2}} + 1} \right)}} = 0 \cr} \)
Câu 42. 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. \(\mathop {\lim }\limits_{x \to – \infty } {{\sqrt {{x^4} + 4} } \over {x + 4}}\)
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.
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\(\mathop {\lim }\limits_{x \to + \infty } {{{x^4} – {x^3} + 11} \over {2x – 7}}\)
\(=\mathop {\lim }\limits_{x \to + \infty } {{{x^3} – {x^2} + {{11} \over x}} \over {2 – {7 \over x}}} = + \infty \)
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 \(\mathop {\lim }\limits_{x \to – \infty } {{\sqrt {{x^4} + 4} } \over {x + 4}} = – \infty \)
Câu 43. Tìm các giới hạn sau :
a. \(\mathop {\lim }\limits_{x \to – \sqrt 3 } {{{x^3} + 3\sqrt 3 } \over {3 – {x^2}}}\)
b. \(\mathop {\lim }\limits_{x \to 4} {{\sqrt x – 2} \over {{x^2} – 4x}}\)
c. \(\mathop {\lim }\limits_{x \to {1^ + }} {{\sqrt {x – 1} } \over {{x^2} – x}}\)
d. \(\mathop {\lim }\limits_{x \to 0} {{\sqrt {{x^2} + x + 1} – 1} \over {3x}}\)
a. Ta có: \({{{x^3} + 3\sqrt 3 } \over {3 – {x^2}}} = {{\left( {x + \sqrt 3 } \right)\left( {{x^2} – x\sqrt 3 + 3} \right)} \over {\left( {x + \sqrt 3 } \right)\left( {\sqrt 3 – x} \right)}} = {{{x^2} – x\sqrt 3 + 3} \over {\sqrt 3 – x}}\)
với \(\,x \ne – \sqrt 3 \)
Do đó : \(\mathop {\lim }\limits_{x \to – \sqrt 3 } {{{x^3} + 3\sqrt 3 } \over {3 – {x^2}}} =\mathop {\lim }\limits_{x \to – \sqrt 3 } {{{x^2} – x\sqrt 3 + 3} \over {\sqrt 3 – x}}= {9 \over {2\sqrt 3 }} = {{3\sqrt 3 } \over 2}\)
b.
\(\eqalign{
& \mathop {\lim }\limits_{x \to 4} {{\sqrt x – 2} \over {{x^2} – 4x}} = \mathop {\lim }\limits_{x \to 4} {{\sqrt x – 2} \over {x\left( {x – 4} \right)}} \cr
& = \mathop {\lim }\limits_{x \to 4} {1 \over {x\left( {\sqrt x + 2} \right)}} = {1 \over {16}} \cr} \)
c.
\(\eqalign{
& \mathop {\lim }\limits_{x \to {1^ + }} {{\sqrt {x – 1} } \over {{x^2} – x}} = \mathop {\lim }\limits_{x \to {1^ + }} {{\sqrt {x – 1} } \over {x\left( {x – 1} \right)}} \cr
& = \mathop {\lim }\limits_{x \to {1^ + }} {1 \over {x\sqrt {x – 1} }} = + \infty \cr} \)
d.
\(\eqalign{
& \mathop {\lim }\limits_{x \to 0} {{\sqrt {{x^2} + x + 1} – 1} \over {3x}} = \mathop {\lim }\limits_{x \to 0} {{{x^2} + x + 1 – 1} \over {3x(\sqrt {{x^2} + x + 1} + 1)}} \cr
& = {1 \over 3}\mathop {\lim }\limits_{x \to 0} {{x + 1} \over {\sqrt {{x^2} + x + 1} + 1}} = {1 \over 6} \cr} \)
Câu 44. Tìm các giới hạn sau :
a. \(\mathop {\lim }\limits_{x \to – \infty } x\sqrt {{{2{x^3} + x} \over {{x^5} – {x^2} + 3}}} \)
b. \(\mathop {\lim }\limits_{x \to – \infty } {{\left| x \right| + \sqrt {{x^2} + x} } \over {x + 10}}\)
c. \(\mathop {\lim }\limits_{x \to + \infty } {{\sqrt {2{x^4} + {x^2} – 1} } \over {1 – 2x}}\)
d. \(\mathop {\lim }\limits_{x \to – \infty } \left( {\sqrt {2{x^2} + 1} + x} \right)\)
a. Với \(x < 0\), ta có :
\(\eqalign{
& x\sqrt {{{2{x^3} + x} \over {{x^5} – {x^2} + 3}}} = – \left| x \right|\sqrt {{{2{x^3} + x} \over {{x^5} – {x^2} + 3}}} \cr
& = – \sqrt {{{{x^2}\left( {2{x^3} + x} \right)} \over {{x^5} – {x^2} + 3}}} = – \sqrt {{{2 + {1 \over {{x^2}}}} \over {1 – {1 \over {{x^3}}} + {1 \over {{x^5}}}}}} \cr} \)
Do đó : \(\mathop {\lim }\limits_{x \to – \infty } x\sqrt {{{2{x^3} + x} \over {{x^5} – {x^2} + 3}}} = – \sqrt 2 \)
b.
\(\eqalign{
& \mathop {\lim }\limits_{x \to – \infty } {{\left| x \right|+\sqrt {{x^2} + x} } \over {x + 10}} = \mathop {\lim }\limits_{x \to – \infty } {{\left| x \right| + \left| x \right|\sqrt {1 + {1 \over x}} } \over {x + 10}} \cr
& = \mathop {\lim }\limits_{x \to – \infty } {{ – x – x\sqrt {1 + {1 \over x}} } \over {x + 10}} = \mathop {\lim }\limits_{x \to – \infty } {{ – 1 – \sqrt {1 + {1 \over x}} } \over {1 + {{10} \over x}}} \cr &= – 2 \cr} \)
c.
\(\eqalign{
& \mathop {\lim }\limits_{x \to + \infty } {{\sqrt {2{x^4} + {x^2} – 1} } \over {1 – 2x}} = \mathop {\lim }\limits_{x \to + \infty } {{{x^2}\sqrt {2 + {1 \over {{x^2}}} – {1 \over {{x^4}}}} } \over {x\left( {{1 \over x} – 2} \right)}} \cr
& = \mathop {\lim }\limits_{x \to + \infty } x{{\sqrt {2 + {1 \over {{x^2}}} – {1 \over {{x^4}}}} } \over {{1 \over x} – 2}} = – \infty \cr
& \text{vì}\,\mathop {\lim }\limits_{x \to + \infty } x = + \infty \,\text{và}\,\mathop {\lim }\limits_{x \to + \infty } {{\sqrt {2 + {1 \over {{x^2}}} – {1 \over {{x^4}}}} } \over {{1 \over x} – 2}} = – {{\sqrt 2 } \over 2} < 0 \cr} \)
d.
\(\eqalign{
& \mathop {\lim }\limits_{x \to – \infty } \left( {\sqrt {2{x^2} + 1} + x} \right) = \mathop {\lim }\limits_{x \to – \infty } {{2{x^2} + x – {x^2}} \over {\sqrt {2{x^2} + x} – x}} \cr
& = \mathop {\lim }\limits_{x \to – \infty } {{x\left( {x + 1} \right)} \over { – x\left( {\sqrt {2 + {1 \over x}} + 1} \right)}} \cr &= \mathop {\lim }\limits_{x \to – \infty } – {{x + 1} \over {\sqrt {2 + {1 \over x} + 1} }} = + \infty \cr
& \text{vì }\,\mathop {\lim }\limits_{x \to – \infty } \left( { – x – 1} \right) = + \infty \cr} \)
Câu 45. Tìm các giới hạn sau :
a. \(\mathop {\lim }\limits_{x \to {0^ + }} {{\sqrt {{x^2} + x} – \sqrt x } \over {{x^2}}}\)
b. \(\mathop {\lim }\limits_{x \to {1^ – }} x{{\sqrt {1 – x} } \over {2\sqrt {1 – x} + 1 – x}}\)
c. \(\mathop {\lim }\limits_{x \to {3^ – }} {{3 – x} \over {\sqrt {27 – {x^3}} }}\)
d. \(\mathop {\lim }\limits_{x \to {2^ + }} {{\sqrt {{x^3} – 8} } \over {{x^2} – 2x}}\)
a.
\(\eqalign{
& \mathop {\lim }\limits_{x \to {0^ + }} {{\sqrt {{x^2} + x} – \sqrt x } \over {{x^2}}} = \mathop {\lim }\limits_{x \to {0^ + }} {x^2 \over {{x^2}\left( {\sqrt {{x^2} + x} + \sqrt x } \right)}} \cr
& = \mathop {\lim }\limits_{x \to {0^ + }} {1 \over {\left( {\sqrt {{x^2} + x} + \sqrt x } \right)}} = + \infty \cr} \)
b. \(\mathop {\lim }\limits_{x \to {1^ – }} x{{\sqrt {1 – x} } \over {2\sqrt {1 – x} + 1 – x}} = \mathop {\lim }\limits_{x \to {1^ – }} {x \over {2 + \sqrt {1 – x} }} = {1 \over 2}\)
c.
\(\eqalign{
& \mathop {\lim }\limits_{x \to {3^ – }} {{3 – x} \over {\sqrt {27 – {x^3}} }} = \mathop {\lim }\limits_{x \to {3^ – }} {{{{\left( {\sqrt {3 – x} } \right)}^2}} \over {\sqrt {\left( {3 – x} \right)\left( {{x^2} + 3x + 9} \right)} }} \cr
& = \mathop {\lim }\limits_{x \to {3^ – }} {{\sqrt {3 – x} } \over {\sqrt {{x^2} + 3x + 9} }} = 0 \cr} \)
d.
\(\eqalign{
& \mathop {\lim }\limits_{x \to {2^ + }} {{\sqrt {{x^3} – 8} } \over {{x^2} – 2x}} = \mathop {\lim }\limits_{x \to {2^ + }} {{\sqrt {\left( {x – 2} \right)\left( {{x^2} + 2x + 4} \right)} } \over {x\left( {x – 2} \right)}} \cr
& = \mathop {\lim }\limits_{x \to {2^ + }} {1 \over x}\sqrt {{{{x^2} + 2x + 4} \over {x – 2}}} = + \infty \cr} \)
Vì
\(\eqalign{
& \mathop {\lim }\limits_{x \to {2^ + }} \sqrt {{x^2} + 2x + 4} = 2\sqrt 3 \cr
& \mathop {\lim }\limits_{x \to {2^ + }} x\sqrt {x – 2} = 0;\,x\sqrt {x – 2} > 0\,\forall x > 2 \cr} \)
