The value of limx→0 ax+bx+cx31/x, is - Infinity Learn by Sri Chaitanya

A
$a b c$
B
$(a b c)^{1 / 3}$
C
$\frac{1}{3} a b c$
D
none of these

Solution:

We have,

$\begin{array}{l} lim_{x \to 0} {\{\frac{a^{x} + b^{x} + c^{x}}{3}\}}^{\frac{1}{x}} \\ = lim_{x \to 0} {\{1 + \frac{a^{x} + b^{x} + c^{x} - 3}{3}\}}^{\frac{1}{x}} \end{array}$

$\begin{array}{l} = lim_{x \to 0} {\{1 + \frac{(a^{x} - 1) + (b^{x} - 1) + (c^{x} - 1)}{3}\}}^{\frac{1}{x}} \\ = e^{lim_{x \to 0} \frac{a^{x}}{3 x} + \frac{b^{x} - 1}{3 x} + \frac{c^{x} - 1}{3 x}} \\ = e^{\frac{1}{3} \{lim_{x \to 0} \frac{a^{x} - 1}{x} + lim_{x \to 0} \frac{b^{x} - 1}{x} + lim_{x \to 0} c^{x} - 1} x^{2}\} \\ = e^{\frac{1}{3} {\log a + \log b + \log c}} \\ = e^{\log (a b c)^{1 / 3}} = (a b c)^{1 / 3} \end{array}$

The value of $lim_{x \to 0} {\{\frac{a^{x} + b^{x} + c^{x}}{3}\}}^{1 / x},$ is

Solution:

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