Functions Question 177
Question: The value of $ \underset{x\to 0}{\mathop{\lim }},{{( \frac{a^{x}+b^{x}+c^{x}}{3} )}^{2/x}} $ ; $ (a,\ b,\ c>0) $ is
Options:
A) $ {{(abc)}^{3}} $
B) $ abc $
C) $ {{(abc)}^{1/3}} $
D) None of these
Show Answer
Answer:
Correct Answer: D
Solution:
Let $ y=\underset{x\to 0}{\mathop{\lim }}{{( \frac{a^{x}+b^{x}+c^{x}}{3} )}^{2/x}} $ $ =\underset{h\to 0}{\mathop{\lim }},h,\sin 1/h=0\times (-1\le \sin 1/h\le 1)=0 $ $ =2,\underset{x\to 0}{\mathop{\lim }}\frac{\log ,(a^{x}+b^{x}+c^{x})-\log 3}{x} $ Now applying L-Hospital?s rule, we have $ \log y=\log ,{{(abc)}^{2/3}},\Rightarrow y={{(abc)}^{2/3}} $
BETA
