Functions Question 192
Question: $ \underset{x\to 0}{\mathop{\lim }},{{( \frac{1+5x^{2}}{1+3x^{2}} )}^{1/x^{2}}}= $
[IIT 1996; DCE 2001]
Options:
A) $ e^{2} $
B) $ e $
C) $ {e^{-2}} $
D) $ {e^{-1}} $
Show Answer
Answer:
Correct Answer: A
Solution:
$ \underset{x\to 0}{\mathop{\lim }}{{( \frac{1+5x^{2}}{1+3x^{2}} )}^{1/x^{2}}}=\frac{\underset{x\to 0}{\mathop{\lim }},{{[ {{(1+5x^{2})}^{1/5x^{2}}} ]}^{5}}}{\underset{x\to 0}{\mathop{\lim }},{{[ {{(1+3x^{2})}^{1/3x^{2}}} ]}^{3}}}=\frac{e^{5}}{e^{3}}=e^{2} $ . $ [\because ,\underset{x\to 0}{\mathop{\lim }}{{(1+x)}^{1/x}}=e] $
BETA
