Bài 2.5 trang 163 Sách bài tập (SBT) Đại số và giải tích 11

Tìm giới hạn của các hàm số sau :

a) \(f\left( x \right) = {{{x^2} - 2x - 3} \over {x - 1}}\) khi \(x \to 3\) ;

b) \(h\left( x \right) = {{2{x^3} + 15} \over {{{\left( {x + 2} \right)}^2}}}\) khi \(x \to  - 2\) ;

c) \(k\left( x \right) = \sqrt {4{x^2} - x + 1} \) khi \(x \to  - \infty \) ;

d) \(f\left( x \right) = {x^3} + {x^2} + 1\) khi \(x \to  - \infty \)  ;

e) \(h\left( x \right) = {{x - 15} \over {x + 2}}\) khi \(x \to  - {2^ + }\) và khi \(x \to  - {2^ - }\)

Giải :

a) 0 ;                            b) \( - \infty \) ;

c) 

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

d) \(\mathop {\lim }\limits_{x \to  - \infty } \left( {{x^3} + {x^2} + 1} \right) = \mathop {\lim }\limits_{x \to  - \infty } {x^3}\left( {1 + {1 \over x} + {1 \over {{x^3}}}} \right) =  - \infty \)

e) \( - \infty \) và \( + \infty \)

Bài 2.6 trang 163 Sách bài tập (SBT) Đại số và giải tích 11

Tính các giới hạn sau :

a) \(\mathop {\lim }\limits_{x \to  - 3} {{x + 3} \over {{x^2} + 2x - 3}}\) ;                            b) \(\mathop {\lim }\limits_{x \to 0} {{{{\left( {1 + x} \right)}^3} - 1} \over x}\) ;

c) \(\mathop {\lim }\limits_{x \to  + \infty } {{x - 1} \over {{x^2} - 1}}\) ;                                d) \(\mathop {\lim }\limits_{x \to 5} {{x - 5} \over {\sqrt x  - \sqrt 5 }}\) ;

e) \(\mathop {\lim }\limits_{x \to  + \infty }  = {{x - 5} \over {\sqrt x  + \sqrt 5 }}\) ;                      f) \(\mathop {\lim }\limits_{x \to  - 2} {{\sqrt {{x^2} + 5}  - 3} \over {x + 2}}\) ;

g) \(\mathop {\lim }\limits_{x \to 1} {{\sqrt x  - 1} \over {\sqrt {x + 3}  - 2}}\) ;                               h) \(\mathop {\lim }\limits_{x \to  + \infty } {{1 - 2x + 3{x^3}} \over {{x^3} - 9}}\) ;

i) \(\mathop {\lim }\limits_{x \to 0} {1 \over {{x^2}}}\left( {{1 \over {{x^2} + 1}} - 1} \right)\) ;                   j) \(\mathop {\lim }\limits_{x \to  - \infty } {{\left( {{x^2} - 1} \right){{\left( {1 - 2x} \right)}^5}} \over {{x^7} + x + 3}}\) ;

Giải:

a) \(\mathop {\lim }\limits_{x \to  - 3} {{x + 3} \over {{x^2} + 2x - 3}} = \mathop {\lim }\limits_{x \to  - 3} {{x + 3} \over {\left( {x - 1} \right)\left( {x + 3} \right)}} = \mathop {\lim }\limits_{x \to  - 3} {1 \over {x - 1}} = {{ - 1} \over 4}\)

b) 

\(\eqalign{
& \mathop {\lim }\limits_{x \to 0} {{{{\left( {1 + x} \right)}^3} - 1} \over x} \cr
& = \mathop {\lim }\limits_{x \to 0} {{\left( {1 + x - 1} \right)\left[ {{{\left( {1 + x} \right)}^2} + \left( {1 + x} \right) + 1} \right]} \over x} \cr
& = \mathop {\lim }\limits_{x \to 0} {{x\left[ {{{\left( {1 + x} \right)}^2} + \left( {1 + x} \right) + 1} \right]} \over x} \cr
& = \mathop {\lim }\limits_{x \to 0} \left[ {{{\left( {1 + x} \right)}^2} + \left( {1 + x} \right) + 1} \right] = 3 \cr} \)

c) \(\mathop {\lim }\limits_{x \to  + \infty } {{x - 1} \over {{x^2} - 1}} = \mathop {\lim }\limits_{x \to  + \infty } {{{1 \over x} - {1 \over {{x^2}}}} \over {1 - {1 \over {{x^2}}}}} = 0\)

d) \(\mathop {\lim }\limits_{x \to 5} {{x - 5} \over {\sqrt x  - \sqrt 5 }}\)

\(= \mathop {\lim }\limits_{x \to 5} {{\left( {\sqrt x  - \sqrt 5 } \right)\left( {\sqrt x  + \sqrt 5 } \right)} \over {\sqrt x  - \sqrt 5 }}\)

\(= \mathop {\lim }\limits_{x \to 5} \left( {\sqrt x  + \sqrt 5 } \right) = 2\sqrt 5 \)

e)

\(\eqalign{
& \mathop {\lim }\limits_{x \to + \infty } {{x - 5} \over {\sqrt x + \sqrt 5 }} \cr
& = \mathop {\lim }\limits_{x \to + \infty } {{1 - {5 \over x}} \over {{1 \over {\sqrt x }} + {{\sqrt 5 } \over x}}} = + \infty \cr} \)  

(Vì \({1 \over {\sqrt x }} + {{\sqrt 5 } \over x} > 0\) với mọi \(x > 0\) ).

f) 

\(\eqalign{
& \mathop {\lim }\limits_{x \to - 2} {{\sqrt {{x^2} + 5} - 3} \over {x + 2}} \cr
& = \mathop {\lim }\limits_{x \to - 2} {{{x^2} + 5 - 9} \over {\left( {x + 2} \right)\left( {\sqrt {{x^2} + 5} + 3} \right)}} \cr
& = \mathop {\lim }\limits_{x \to - 2} {{\left( {x - 2} \right)\left( {x + 2} \right)} \over {\left( {x + 2} \right)\left( {\sqrt {{x^2} + 5} + 3} \right)}} \cr
& = \mathop {\lim }\limits_{x \to - 2} {{x - 2} \over {\sqrt {{x^2} + 5} + 3}} = {{ - 2} \over 3} \cr} \)

g) 

\(\eqalign{
& \mathop {\lim }\limits_{x \to 1} {{\sqrt x - 1} \over {\sqrt {x + 3} - 2}} \cr
& = \mathop {\lim }\limits_{x \to 1} {{\left( {\sqrt x - 1} \right)\left( {\sqrt {x + 3} + 2} \right)} \over {x + 3 - 4}} \cr
& = \mathop {\lim }\limits_{x \to 1} {{\left( {\sqrt {x - 1} } \right)\left( {\sqrt {x + 3} + 2} \right)} \over {x - 1}} \cr
& = \mathop {\lim }\limits_{x \to 1} {{\left( {\sqrt x - 1} \right)\left( {\sqrt {x + 3} + 2} \right)} \over {\left( {\sqrt x - 1} \right)\left( {\sqrt x + 1} \right)}} \cr
& = \mathop {\lim }\limits_{x \to 1} {{\sqrt {x + 3} + 2} \over {\sqrt x + 1}} = 2 \cr} \)

h) \(\mathop {\lim }\limits_{x \to  + \infty } {{1 - 2x + 3{x^3}} \over {{x^3} - 9}} = \mathop {\lim }\limits_{x \to  + \infty } {{{1 \over {{x^3}}} - {2 \over {{x^2}}} + 3} \over {1 - {9 \over {{x^3}}}}} = 3\)

i) 

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

j)

\(\eqalign{
& \mathop {\lim }\limits_{x \to - \infty } {{\left( {{x^2} - 1} \right){{\left( {1 - 2x} \right)}^5}} \over {{x^7} + x + 3}} \cr
& = \mathop {\lim }\limits_{x \to - \infty } {{{x^2}\left( {1 - {1 \over {{x^2}}}} \right).{x^5}{{\left( {{1 \over x} - 2} \right)}^5}} \over {{x^7} + x + 3}} \cr
& = \mathop {\lim }\limits_{x \to - \infty } {{\left( {1 - {1 \over {{x^2}}}} \right){{\left( {{1 \over x} - 2} \right)}^5}} \over {1 + {1 \over {{x^6}}} + {3 \over {{x^7}}}}} \cr
& = {\left( { - 2} \right)^5} = - 32 \cr}\)

Bài 2.7 trang 164 Sách bài tập (SBT) Đại số và giải tích 11

Tính giới hạn của các hàm số sau khi \(x \to  + \infty \) và khi \(x \to  - \infty \)

a) \(f\left( x \right) = {{\sqrt {{x^2} - 3x} } \over {x + 2}}\) ;

b) \(f\left( x \right) = x + \sqrt {{x^2} - x + 1}\) ;

c) \(f\left( x \right) = \sqrt {{x^2} - x}  - \sqrt {{x^2} + 1} \) .

Giải:

a) Khi  \(x \to  + \infty \)

\(\eqalign{
& \mathop {\lim }\limits_{x \to + \infty } {{\sqrt {{x^2} - 3x} } \over {x + 2}} = \mathop {\lim }\limits_{x \to + \infty } {{\left| x \right|\sqrt {1 - {3 \over x}} } \over {x + 2}} \cr
& = \mathop {\lim }\limits_{x \to + \infty } {{x\sqrt {1 - {3 \over x}} } \over {x + 2}} = \mathop {\lim }\limits_{x \to + \infty } {{\sqrt {1 - {3 \over x}} } \over {1 + {2 \over x}}} = 1 \cr} \)

Khi \(x \to  - \infty \)

\(\eqalign{
& \mathop {\lim }\limits_{x \to - \infty } {{\sqrt {{x^2} - 3x} } \over {x + 2}} = \mathop {\lim }\limits_{x \to - \infty } {{\left| x \right|\sqrt {1 - {3 \over x}} } \over {x + 2}} \cr
& = \mathop {\lim }\limits_{x \to - \infty } {{ - x\sqrt {1 - {3 \over x}} } \over {x + 2}} = \mathop {\lim }\limits_{x \to - \infty } {{ - \sqrt {1 - {3 \over x}} } \over {1 + {2 \over x}}} = - 1 \cr}\) ;

b)  Khi  \(x \to  + \infty \)

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

Khi \(x \to  - \infty \)

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

c) Khi  \(x \to  + \infty \)

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

Khi \(x \to  - \infty \)

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

Bài 2.8 trang 164 Sách bài tập (SBT) Đại số và giải tích 11

Cho hàm số \(f\left( x \right) = {{2{x^2} - 15x + 12} \over {{x^2} - 5x + 4}}\) có đồ thị như hình 4 

a)      Dựa vào đồ thị, dự đoán giới hạn của hàm \(f\left( x \right)\) số khi \(x \to {1^ + }{\rm{ }};{\rm{ }}x \to {1^ - }{\rm{ }};{\rm{ }}x \to {4^ + }{\rm{ }};{\rm{ }}x \to {4^ - }{\rm{ }};{\rm{ }}x \to  + \infty {\rm{ }};{\rm{ }}x \to  - \infty \)

b)      Chứng minh dự đoán trên.

Giải:

a)      Dự đoán :

\(\eqalign{
& \mathop {\lim }\limits_{x \to {1^ + }} f\left( x \right) = + \infty {\rm{ ; }}\mathop {\lim }\limits_{x \to {1^ - }} f\left( x \right) = - \infty {\rm{ ; }}\mathop {\lim }\limits_{x \to {4^ + }} f\left( x \right) = - \infty {\rm{ ;}} \cr
& {\rm{ }}\mathop {\lim }\limits_{x \to {4^ - }} f\left( x \right) = + \infty {\rm{ ;}}\mathop {\lim }\limits_{x \to + \infty } f\left( x \right) = 2{\rm{ ; }}\mathop {\lim }\limits_{x \to - \infty } f\left( x \right) = 2. \cr} \) 

b)      Ta có

\(\mathop {\lim }\limits_{x \to {1^ + }} \left( {2{x^2} - 15x + 12} \right) =  - 1 < 0,{\rm{  }}\mathop {\lim }\limits_{x \to {1^ + }} \left( {{x^2} - 5x + 4} \right) = 0\)

và \({x^2} - 5x + 4 < 0\) với mọi \(x \in \left( {1;4} \right)\) nên \(\mathop {\lim }\limits_{x \to {1^ + }} {{2{x^2} - 15x + 12} \over {{x^2} - 5x + 4}} =  + \infty \)

Vì

\(\eqalign{
& \mathop {\lim }\limits_{x \to {1^ - }} \left( {2{x^2} - 15x + 12} \right) = - 1 < 0, \cr
& \mathop {\lim }\limits_{x \to {1^ - }} \left( {{x^2} - 5x + 4} \right) = 0 \cr} \)

và \({x^2} - 5x + 4 > 0\) với mọi x < 1 nên \(\mathop {\lim }\limits_{x \to {1^ - }} {{2{x^2} - 15x + 12} \over {{x^2} - 5x + 4}} =  - \infty \)

Vì

\(\eqalign{
& \mathop {\lim }\limits_{x \to {4^ + }} \left( {2{x^2} - 15x + 12} \right) = - 16 < 0, \cr
& \mathop {\lim }\limits_{x \to {4^ + }} \left( {{x^2} - 5x + 4} \right) = 0 \cr} \)

và \({x^2} - 5x + 4 > 0\) với mọi x > 4 nên \(\mathop {\lim }\limits_{x \to {4^ + }} {{2{x^2} - 15x + 12} \over {{x^2} - 5x + 4}} =  - \infty \)

Vì

\(\eqalign{
& \mathop {\lim }\limits_{x \to {4^ - }} \left( {2{x^2} - 15x + 12} \right) = - 16 < 0, \cr
& \mathop {\lim }\limits_{x \to {4^ - }} \left( {{x^2} - 5x + 4} \right) = 0 \cr} \)

và \({x^2} - 5x + 4 < 0\) với mọi \(x \in \left( {1;4} \right)\) nên \(\mathop {\lim }\limits_{x \to {4^ - }} {{2{x^2} - 15x + 12} \over {{x^2} - 5x + 4}} =  + \infty\) ;

\(\mathop {\lim }\limits_{x \to  + \infty } {{2{x^2} - 15x + 12} \over {{x^2} - 5x + 4}} = \mathop {\lim }\limits_{x \to  + \infty } {{2 - {{15} \over x} + {{12} \over {{x^2}}}} \over {1 - {5 \over x} + {4 \over {{x^2}}}}} = 2\)

\(\mathop {\lim }\limits_{x \to  - \infty } {{2{x^2} - 15x + 12} \over {{x^2} - 5x + 4}} = \mathop {\lim }\limits_{x \to  - \infty } {{2 - {{15} \over x} + {{12} \over {{x^2}}}} \over {1 - {5 \over x} + {4 \over {{x^2}}}}} = 2\)

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