Evaluate: _ x 0 ( a^x + b^x + c^x 3 )^ 2 x - Vidyadip

MCQ

Evaluate: $\displaystyle \lim_{\text{x} \rightarrow 0}{\left( \frac{\text{a}^\text{x} + \text{b}^\text{x} + \text{c}^\text{x}}{3} \right)^{\frac{2}{\text{x}}}}$

A
$a + b + c$
B
$(\text{abc})^{\frac{2}{4}}$
C
$(\text{abc})^2$
✓
$(\text{abc})^{\frac{2}{3}}$

✓

Answer

Correct option: D.

$(\text{abc})^{\frac{2}{3}}$

$ \displaystyle \lim _{ \text{x} \rightarrow 0 }{ { \left( \frac { { \text{a} }^{ \text{x} }+{ \text{b} }^{ \text{x} }+{ \text{c} }^{ \text{x} } }{ 3 } \right) }^{ \frac{ 2 }{ \text{x} } } } ={ \left( \frac { 3 }{ 3 } \right) }^{\frac { 2 }{ 0 } }$
$ =1\infty \text{ form}={ \text{e} }^{ \displaystyle \lim _{ \text{x}\rightarrow 0 }{ { \left( \cfrac { { \text{a} }^{ \text{x} }+{ \text{b} }^{ \text{x} }+{ \text{c} }^{ \text{x} } }{ 3 } -1 \right) }^{\frac{ 2 }{ \text{x} } } } }$
$ =\text{e}\frac{2}{3}(\log \text{a}+\log \text{b}+\log \text{c})$
$= (\text{abc})^{\frac{2}{3}}$

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