Question

Evaluate: $\underset{x\to 0}{lim}{\left(\frac{a^x+b^x+c^x}{3}\right)}^{2/x}$

• A. $a+b+c$
• B. $(abc{)}^{1/3}$
• C. $(abc{)}^2$
• D. $(abc{)}^{2/3}$

Answer 1

Answer: (D)

$(abc{)}^{2/3}$

$\underset{x\to 0}{lim}{\left(\frac{a^x+b^x+c^x}{3}\right)}^{2/x}={\left(\frac{3}{3}\right)}^{2/0}$

$=1^{\infty }$form

$=e^{\underset{x\to 0}{lim}{(\frac{a^x+b^x+c^x}{3}-1)}^{2/x}}$

$=e^{\underset{x\to 0}{lim}\frac{2}{3}(\frac{a^x-1}{x}+\frac{b^x-1}{x}+\frac{c^x-1}{x})}$

$=e^{\frac{2}{3}(\log a+\log b+\log c)}$

$=e^{{\log }_e{\left(abc\right)}^{2/3}}$
$={\left(abc\right)}^{2/3}$