Differentiation Question 162
Question: $ \underset{x\to \infty }{\mathop{\lim }}{{( \frac{x^{2}+5x+3}{x^{2}+x+2} )}^{x}} $
Options:
A) $ e^{4} $
B) $ e^{2} $
C) $ e^{3} $
D) 1
Show Answer
Answer:
Correct Answer: A
Solution:
[a] $ \underset{x\to \infty }{\mathop{\lim }}{{( \frac{x^{2}+5x+3}{x^{2}+x+2} )}^{x}} $
$ =\underset{x\to \infty }{\mathop{\lim }}{{( 1+\frac{4x+1}{x^{2}+x+2} )}^{x}} $
$ =\underset{x\to \infty }{\mathop{\lim }}{{[ {{( 1+\frac{4x+1}{x^{2}+x+2} )}^{\frac{x^{2}+x+2}{4x+1}}} ]}^{\frac{(4x+1)x}{x^{2}+x+2}}} $
$ =\underset{e^{x}\to \infty }{\mathop{\lim }}\frac{4x^{2}+x}{x^{2}+x+2}={e^{\underset{x\to \infty }{\mathop{\lim }}\frac{4+\frac{1}{x}}{1+\frac{1}{x}+\frac{2}{x^{2}}}}}=e^{4} $
BETA
