Let a1 a2 a3- an1- p1 p2 p3- pn0 be such that p1-p2-p3- - -pn-1- If Fx-p1a1x-p2a2x- - -pnanx1-x- then

Lim_x→0F(x)=lim_x→0(p_1a_1^x+p_2a_2^x+ ⋯ +p_na_n^x)^1/x =e^lim_x→0(p_1a_1^x+p_2a_2^x+ ⋯ +p_na_n^x 1x) =e^lim_x→0(p_1a_1^xln a_1+p_2a_2^xln a_2+ ⋯ +p_na_n^xln a ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XIII

Mathematics

Indeterminate Forms

Let a1 a2 a3…...

Question

Let $a_{1} > a_{2} > a_{3} > \dots > a_{n} > 1;$ $p_{1} > p_{2} > p_{3} > \dots > p_{n} > 0$ be such that $p_{1} + p_{2} + p_{3} + \dots + p_{n} = 1.$ If $F (x) = {(p_{1} a_{1}^{x} + p_{2} a_{2}^{x} + \dots + p_{n} a_{n}^{x})}_{1}^{1 / x},$ then

lim x \to 0 F (x) = a_{1}^{p_{1}} a_{2}^{p_{2}} \dots a_{n}^{p_{n}}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

lim x \to 0 F (x) = a_{1}^{p_{1}} + a_{2}^{p_{2}} + \dots + a_{n}^{p_{n}}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

lim x \to \infty F (x) = a_{1}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

lim x \to - \infty F (x) = a_{n}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

Open in App

Solution

The correct option is D $lim x \to - \infty F (x) = a_{n}$
$lim x \to 0 F (x) = lim x \to 0 {(p_{1} a_{1}^{x} + p_{2} a_{2}^{x} + \dots + p_{n} a_{n}^{x})}_{1}^{1 / x}$
$= e^{lim x \to 0 ⎛ ⎜ ⎝ \frac{p_{1} a_{1}^{x} + p_{2} a_{2}^{x} + \dots + p_{n} a_{n}^{x} - 1}{x} ⎞ ⎟ ⎠}$
$= e^{lim x \to 0 (p_{1} a_{1}^{x} ln a_{1} + p_{2} a_{2}^{x} ln a_{2} + \dots + p_{n} a_{n}^{x} ln a_{n})}$
$= e^{(p_{1} ln a_{1} + p_{2} ln a_{2} + \dots + p_{n} ln a_{n})}$
$= e^{(ln a_{1}^{p_{1}} + ln a_{2}^{p_{2}} + \dots + ln a_{n}^{p_{n}})}$
$= e^{(ln a_{1}^{p_{1}} a_{2}^{p_{2}} \dots a_{n}^{p_{n}})}$
$= a_{1}^{p_{1}} a_{2}^{p_{2}} \dots a_{n}^{p_{n}}$

$L = lim x \to \infty F (x) = lim x \to \infty {(p_{1} a_{1}^{x} + p_{2} a_{2}^{x} + \dots + p_{n} a_{n}^{x})}_{1}^{1 / x} (\infty^{0} form)$
$\Rightarrow ln L = lim x \to \infty \frac{ln (p_{1} a_{1}^{x} + p_{2} a_{2}^{x} + \dots + p_{n} a_{n}^{x})}{x}$
Using L'Hopitals' rule, we get
$ln L = lim x \to \infty \frac{p_{1} a_{1}^{x} ln a_{1} + p_{2} a_{2}^{x} ln a_{2} + \dots + p_{n} a_{n}^{x} ln a_{n}}{p_{1} a_{1}^{x} + p_{2} a_{2}^{x} + \dots + p_{n} a_{n}^{x}}$

Multiplying and dividing by $a_{1}^{x}$ in RHS, we get
${(\frac{a_{2}}{a_{1}})}^{x}, {(\frac{a_{3}}{a_{1}})}^{x},$ etc. all vanish as $x \to \infty$ because $a_{1} > a_{2} > a_{3} > \dots a_{n} > 1$
$\Rightarrow ln L = \frac{p_{1} ln a_{1}}{p_{1}} = ln a_{1}$
Hence, $ln L = ln a_{1}$
$\Rightarrow L = a_{1}$

Let $lim x \to - \infty F (x) = L_{1}$
$∴ ln L_{1} = lim x \to \infty \frac{p_{1} a_{1}^{x} ln a_{1} + p_{2} a_{2}^{x} ln a_{2} + \dots + p_{n} a_{n}^{x} ln a_{n}}{p_{1} a_{1}^{x} + p_{2} a_{2}^{x} + \dots + p_{n} a_{n}^{x}}$
Multiplying and dividing by $(a_{n})^{x}$ in RHS ${(\frac{a_{1}}{a_{n}})}^{x}, {(\frac{a_{2}}{a_{n}})}^{x},$ etc. vanish as $x \to - \infty$
$\Rightarrow ln L_{1} = \frac{p_{n} ln a_{n}}{p_{n}}$
$\Rightarrow L_{1} = a_{n}$

Suggest Corrections

Similar questions

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 - \infty F (x)

equals

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 \infty} F (x)

equals

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

Q. Let

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

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

be such that

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},

then

Q. Let

f (x)

be an invertible function such that

f^{'} (x) > 0

and

f^{''} (x) > 0

for all

x \in R,

then which of the following is/are correct ?
(where

x_{1}, x_{2}, \dots, x_{n}

are different points)

Join BYJU'S Learning Program

Let a1>a2>a3>…>an>1; p1>p2>p3>⋯>pn>0 be such that p1+p2+p3+⋯+pn=1. If F(x)=(p1ax1+p2ax2+⋯+pnaxn)1/x, then

Let $a_{1} > a_{2} > a_{3} > \dots > a_{n} > 1;$ $p_{1} > p_{2} > p_{3} > \dots > p_{n} > 0$ be such that $p_{1} + p_{2} + p_{3} + \dots + p_{n} = 1.$ If $F (x) = {(p_{1} a_{1}^{x} + p_{2} a_{2}^{x} + \dots + p_{n} a_{n}^{x})}_{1}^{1 / x},$ then