A) \[0\]
B) \[1\]
C) \[{{e}^{2}}\]
D) \[{{e}^{4}}\]
Correct Answer: C
Solution :
\[\underset{x\to \infty }{\mathop{\lim }}\,{{\left( \frac{x+3}{x+1} \right)}^{x+2}}\] \[=\underset{x\to \infty }{\mathop{\lim }}\,{{\left( 1+\frac{2}{x+1} \right)}^{\left( \frac{x+1}{2} \right)\times \frac{2}{(x+1)}\times (x+2)}}\] \[=\underset{x\to \infty }{\mathop{\lim }}\,{{\left( 1+\frac{2}{x+1} \right)}^{\left( \frac{x+1}{2} \right)\frac{2(x+2)}{x+1}}}\] \[=\underset{x\to \infty }{\mathop{\lim }}\,{{\left[ {{\left( 1+\frac{2}{x+1} \right)}^{\frac{x+1}{2}}} \right]}^{\frac{2\left( 1+\frac{2}{x} \right)}{\left( 1+\frac{1}{x} \right)}}}\] \[={{e}^{2}}\]
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