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Câu 41 trang 166 SGK Đại số và Giải tích 11 Nâng cao

Tìm các giới hạn sau :
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Tìm các giới hạn sau :

LG a

\(\mathop {\lim }\limits_{x \to + \infty } \left( {\sqrt {{x^2} + 1} - x} \right)\)

Phương pháp giải:

Nhân và chia với biểu thức \(\left( {\sqrt {{x^2} + 1} +x} \right)\)

Lời giải chi tiết:

Dạng ∞ - ∞

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

LG b

\(\mathop {\lim }\limits_{x \to 1} {{\sqrt {2x - {x^2}} - 1} \over {{x^2} - x}}\)

Phương pháp giải:

Nhân cả tử và mẫu với biểu thức \(\left( {\sqrt {2x - {x^2}} + 1} \right)\)

Lời giải chi tiết:

Dạng  \({0 \over 0}\)

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

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