Cho đa thức P(x) thỏa mãn lim x → 3 P ( x ) − 2 x − 3 = 2 . Tính lim x → 3 P ( x ) − 2 ( x 2 − 9 ) ( √ P ( x ) + 2 + 1 )

2/29

Cho đa thức P(x) thỏa mãn \[\mathop {\lim }\limits_{{\rm{x}} \to 3} \frac{{{\rm{P}}\left( {\rm{x}} \right) - 2}}{{{\rm{x}} - 3}} = 2\]. Tính \[\mathop {\lim }\limits_{{\rm{x}} \to 3} \frac{{{\rm{P}}\left( {\rm{x}} \right) - 2}}{{\left( {{{\rm{x}}^2} - 9} \right)\left( {\sqrt {{\rm{P}}\left( {\rm{x}} \right) + 2} + 1} \right)}}\]

\[\frac{1}{3}\]

\[\frac{1}{6}\]

\[\frac{1}{9}\]

\[\frac{2}{3}\]

Giải thích

Vì\[\mathop {\lim }\limits_{{\rm{x}} \to 3} \frac{{{\rm{P}}\left( {\rm{x}} \right) - 2}}{{{\rm{x}} - 3}} = 2 \Rightarrow {\rm{P}}\left( 3 \right) - 2 = 0 \Rightarrow {\rm{P}}\left( {\rm{3}} \right) = 2\]

Ta có:\[\mathop {\lim }\limits_{{\rm{x}} \to 3} \frac{{{\rm{P}}\left( {\rm{x}} \right) - 2}}{{\left( {{{\rm{x}}^2} - 9} \right)\left( {\sqrt {{\rm{P}}\left( {\rm{x}} \right) + 2} + 1} \right)}} = \mathop {\lim }\limits_{{\rm{x}} \to 3} \frac{{{\rm{P}}\left( {\rm{x}} \right) - 2}}{{\left( {{\rm{x}} - 3} \right)}}.\frac{1}{{\left( {{\rm{x}} + 3} \right)\left( {\sqrt {{\rm{P}}\left( {\rm{x}} \right) + 2} + 1} \right)}}\]

\[ = \mathop {\lim }\limits_{{\rm{x}} \to 3} \frac{{{\rm{P}}\left( {\rm{x}} \right) - 2}}{{\left( {{\rm{x}} - 3} \right)}}.\mathop {\lim }\limits_{{\rm{x}} \to 3} \frac{1}{{\left( {{\rm{x}} + 3} \right)\left( {\sqrt {{\rm{P}}\left( {\rm{x}} \right) + 2} + 1} \right)}} = 2.\frac{1}{{\left( {3 + 3} \right)\left( {\sqrt {2 + 2} + 1} \right)}} = \frac{1}{9}\]

Chọn đáp án C

Đáp án cần chọn là: C