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Properties of Composite Function

Let a1 > a2...

Question

Let $a_{1} > a_{2} > a_{3} > . . . . > a_{n} > 1.$
$p_{1} > p_{2} > p_{3} > . . . . > p_{n} > 0$ such that $p_{1} + p_{2} + p_{3} + . . . + p_{n} = 1.$
Also, $F (x) = (p_{1} a_{1}^{x} + p_{2} a_{2}^{x} + . . . + p_{n} a_{n}^{x})^{1 / x}$
${lim}_{x \to 0^{+}} F (x)$ equals

p_{1} l n a_{1} + p_{2} l i n a_{2} + . . . + p_{n} l n a_{n}

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a_{1}^{p_{1}} + a_{2}^{p_{2}} + . . . . . + a_{n}^{p_{n}}

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a_{1}^{p_{1}} \cdot a_{2}^{p_{2}} . . . . a_{n}^{p_{n}}

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\sum_{r = 1}^{n} a_{r} p_{r}

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Solution

The correct option is D $a_{1}^{p_{1}} \cdot a_{2}^{p_{2}} . . . . a_{n}^{p_{n}}$
${lim}_{x \to 0^{+}} F (x) = {lim}_{x \to 0^{+}} (p_{1} a_{1}^{x} + p_{2} a_{2}^{x} + . . . + p_{n} a_{n}^{x})^{1 / x}$ (1 $^{\infty}$ form)
$= e^{{lim}_{x \to 0} ⎛ ⎜ ⎝ \frac{p_{1} a_{1}^{x} + p_{2} a_{2}^{x} + . . . + p_{n} a_{n}^{x} - 1}{x} ⎞ ⎟ ⎠}$
$= e^{{lim}_{x \to 0} (p_{1} a_{1}^{x} l n a_{1} + p_{2} a_{2}^{x} l n a_{2} + . . . + p_{n} a_{n}^{x} l n a_{n})}$
$= e^{(p_{1} l n a_{1} + p_{2} l n a_{2} + . . . + p_{n} l n a_{n})}$
$= e^{(l n a_{1}^{p_{1}} + l n a_{2}^{p_{2}} + . . . + l n a_{n}^{p_{n}})}$
$= e^{l n a_{1}^{p_{1}} a_{2}^{p_{2}} . . . . a_{n}^{p_{n}}}$
$= a_{1}^{p_{1}} \cdot a_{2}^{p_{2}} a_{3}^{p_{3}} . . . . a_{n}^{p_{n}}$
Hence, option 'C' is correct.

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Similar questions

a_{1} > a_{2} > a_{3} > \dots a_{n} > 1;

p_{1} > p_{2} > p_{3} > . . . p_{n} > 0

p_{1} + p_{2} + p_{3} + \dots + p_{n} = 1.

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a_{1} > a_{2} > a_{3} > . . . . > a_{n} > 1.

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