A) \[0\]
B) \[1\]
C) \[\infty \]
D) \[-\infty \]
Correct Answer: A
Solution :
\[\underset{x\to \infty }{\mathop{\lim }}\,\frac{{{2}^{-n}}({{n}^{2}}+5n+6)}{(n+4)(n+5)}\] \[=\underset{x\to \infty }{\mathop{\lim }}\,\frac{{{n}^{2}}\left( 1+\frac{5}{n}+\frac{6}{{{n}^{2}}} \right)}{{{2}^{n}}.{{n}^{2}}\left( 1+\frac{4}{n} \right)\left( 1+\frac{5}{n} \right)}\] \[=\underset{x\to \infty }{\mathop{\lim }}\,\frac{{{n}^{2}}\left( 1+\frac{5}{n}+\frac{6}{{{n}^{2}}} \right)}{{{2}^{n}}\left( 1+\frac{4}{n} \right)\left( 1+\frac{5}{n} \right)}=0\]
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