Let P n - a p n-1 -1- - n -2-3- and let P 1 - a x -1 where a - R - then evaluate lim x- 0 P n x

P _ 1 =( a ^ x 1) P _ 2 =( a ^ P _ 1 1)=( a ^ ( a ^ x 1) 1) P _ 3 =( a ^ P _ 2 1)=( a ^ ( a ^ ( a ^ x 1) 1) 1) We know that x→ 0 lim ...

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Indeterminate Forms

Let P n = a...

Question

Let $P_{n} = a^{p_{n - 1}} - 1, \forall n = 2, 3, . . . .$ and let $P_{1} = a^{x} - 1$ where $a \in R^{+}$ then evaluate $lim x \to 0 \frac{P_{n}}{x}$

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

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

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

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)

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

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

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

F (x) = (p_{1} a_{1}^{x} + p_{2} a_{2}^{x} + . . . + p_{n} a_{n}^{x})^{1 / x}

lim x \to - \infty F (x)

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

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

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

F (x) = (p_{1} a_{1}^{x} + p_{2} a_{2}^{x} + . . . + p_{n} a_{n}^{x})^{1 / x}

{lim}_{x \to \infty} F (x)

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.

F (x) = {(p_{1} a_{1}^{x} + p_{2} a_{2}^{x} + \dots + p_{n} a_{n}^{x})}_{1}^{1 / x},

P (n)

P (n) = P (n + 1) \forall n \in N

P (n)

n

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