Q122 / 200 · Manipal (MET) 2014 · Mathematics

The value of the $\lim _{x \rightarrow 0}\left(\frac{a^x+b^x+c^x}{3}\right)^{2 / x}$, $(a, b, c\gt0)$ is

  • A
    $(a b c)^3$
  • B
    $abc$
  • C
    $(a b c)^{1 / 3}$
  • Correct answer
    D
    None of these

Solution

$\begin{aligned} & \text { Let } y=\lim _{x \rightarrow 10}\left(\frac{a^x+b^x+c^x}{3}\right)^{2 / x} \\ & \begin{array}{l}\Rightarrow \log y=\lim _{x \rightarrow 10} \frac{2}{x} \log \left(\frac{a^x+b^x+c^x}{3}\right) \\ \quad=\lim _{x \rightarrow 10} \frac{\log \left(a^x+b^x+c^x\right)-\log 3}{x} \\ \Rightarrow \log y=\log (a b c)^{2 / 3} \quad \text { (by L-Hospital's rule) } \\ \Rightarrow \quad y=(a b c)^{2 / 3}\end{array}\end{aligned}$

Hence, the correct answer is option: 'D'.

More Limits Questions