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Question Number 29689 by mrW2 last updated on 11/Feb/18

find lim_(x→0) ((2/(1+a^x )))^(1/x) =?

f i n d \lim_{x \to 0} {(\frac{2}{1 + a^{x}})}^{\frac{1}{x}} = ?

Commented by abdo imad last updated on 13/Feb/18

let put A(x)=((2/(1+a^x )))^(1/x) ⇒ln(A(x))= (1/x)ln( (2/(1+a^x ))) =(1/x)( ln(2)−ln(1+a^x ))=(1/x)(ln(2)−ln(2 +a^x −1)) =(1/x)(−ln(1+((a^x −1)/2)))⇒A(x)= e^(((−1)/x)ln(1 +((a^x −1)/2))) for x∈v(0) ((a^x −1)/2) ∈v(0) and ln(1+u) ∼u ⇒ln(1+((a^x −1)/2))∼((a^x −1)/2)⇒ −(1/x)ln(...) ∼ ((1−a^x )/(2x)) by hospital theorem lim_(x→0) ((1−a^x )/(2x))=lim_(x→0) ((1−e^(xlna) )/(2x))=lim_(x→0) ((−lna e^(xlna) )/2) =ln((1/( (√a)))) ⇒lim_(x→0) A(x)= (1/( (√a))) .

l e t p u t A (x) = {(\frac{2}{1 + a^{x}})}^{\frac{1}{x}} \Rightarrow l n (A (x)) = \frac{1}{x} l n (\frac{2}{1 + a^{x}})

= \frac{1}{x} (l n (2) - l n (1 + a^{x})) = \frac{1}{x} (l n (2) - l n (2 + a^{x} - 1))

= \frac{1}{x} (- l n (1 + \frac{a^{x} - 1}{2})) \Rightarrow A (x) = e^{\frac{- 1}{x} l n (1 + \frac{a^{x} - 1}{2})} f o r x \in v (0)

\frac{a^{x} - 1}{2} \in v (0) a n d l n (1 + u) \sim u \Rightarrow l n (1 + \frac{a^{x} - 1}{2}) \sim \frac{a^{x} - 1}{2} \Rightarrow

- \frac{1}{x} l n (\dots) \sim \frac{1 - a^{x}}{2 x} b y h o s p i t a l t h e o r e m

{l i m}_{x \to 0} \frac{1 - a^{x}}{2 x} = {l i m}_{x \to 0} \frac{1 - e^{x l n a}}{2 x} = {l i m}_{x \to 0} \frac{- l n a e^{x l n a}}{2}

= l n (\frac{1}{\sqrt{a}}) \Rightarrow {l i m}_{x \to 0} A (x) = \frac{1}{\sqrt{a}} .

Commented by mrW2 last updated on 13/Feb/18

thanks sir!

t h a n k s s i r!

Answered by ajfour last updated on 11/Feb/18

let lim_(x→0) ((2/(1+a^x )))^(1/x) = L ln L =lim_(x→0) [((ln (1+((1−a^x )/(1+a^x ))))/x)] =lim_(x→0) [(((((1−a^x )/(1+a^x ))))/x)] =lim_(x→0) (((−xln a−(x^2 /2)(ln a)^2 −...))/(x(1+a^x ))) =lim_(x→0) [−ln a(((1+((xln a)/2)+..)/(1+a^x )))] ⇒ ln L = −((ln a)/2) ⇒ L = (1/( (√a))) .

l e t \lim_{x \to 0} {(\frac{2}{1 + a^{x}})}^{1 / x} = L

\ln L = \lim_{x \to 0} [\frac{\ln (1 + \frac{1 - a^{x}}{1 + a^{x}})}{x}]

= \lim_{x \to 0} [\frac{(\frac{1 - a^{x}}{1 + a^{x}})}{x}]

= \lim_{x \to 0} \frac{(- x \ln a - \frac{x^{2}}{2} {(\ln a)}^{2} - \dots)}{x (1 + a^{x})}

= \lim_{x \to 0} [- \ln a (\frac{1 + \frac{x \ln a}{2} + . .}{1 + a^{x}})]

\Rightarrow \ln L = - \frac{\ln a}{2}

\Rightarrow L = \frac{1}{\sqrt{a}} .

Commented by 803jaideep@gmail.com last updated on 11/Feb/18

are you guys teacher?.... you solve all ques easily..... or a dtudent of +1 +2?

are you guys teacher ? \dots . you solve all ques easily \dots . . or a dtudent of + 1 + 2 ?

Commented by ajfour last updated on 11/Feb/18

i teach physics for 10+1,10+2 . what about you sir ?

i t e a c h p h y s i c s f o r 10 + 1, 10 + 2 . w h a t a b o u t y o u s i r ?

Commented by 803jaideep@gmail.com last updated on 11/Feb/18

im a +1 student

im a + 1 student

Commented by math solver last updated on 11/Feb/18

wow! still solving maths q. efficiently. my phy. teacher should learn then!

wow! still solving maths q . efficiently .

my phy . teacher should learn then!

Commented by math solver last updated on 11/Feb/18

by reading the answers of mrw2 sir i assumed he is a professor at some university but later i came to know he is an engineer and have very little (almost no) use of phy.,maths.

by reading the answers of mrw 2 sir

i assumed he is a professor at some

university but later i came to know

he is an engineer and have very little

(almost no) use of phy ., maths .

Commented by 803jaideep@gmail.com last updated on 11/Feb/18

are u also teacher?

are u also teacher ?

Commented by math solver last updated on 11/Feb/18

no, i am also +1 student. moving to +2 this april.

no, i am also + 1 student .

moving to + 2 this april .

Commented by math solver last updated on 11/Feb/18

just go to question ID 22537.

just go to question ID 22537 .

Commented by mrW2 last updated on 11/Feb/18

Thank you ajfour sir! I guess the answer should be 1/√a, see Q29633, but I had no proof. Now it is sure.

Commented by 803jaideep@gmail.com last updated on 11/Feb/18

is ur target iit?... u f vry intelligent at 16 im sldo 16

is ur target iit ? \dots u f vry intelligent at 16 im sldo 16

Commented by math solver last updated on 11/Feb/18

yes ! btw, i am sorry i am a nerd infront of these guys asking silly questions :(

yes!

btw, i am sorry i am a nerd infront of

these guys asking silly questions : (

Commented by 803jaideep@gmail.com last updated on 11/Feb/18

thn think about me... i also target iit

thn think about me \dots i also target iit

Commented by 803jaideep@gmail.com last updated on 11/Feb/18

fir bhi?...avg?... me abt 6 hrs generally

fir bhi ? \dots avg ? \dots me abt 6 hrs generally

Commented by 803jaideep@gmail.com last updated on 11/Feb/18

im in +1moving +2 this april

im in + 1 moving + 2 this april

Commented by 803jaideep@gmail.com last updated on 11/Feb/18

how many hours a day u study?

how many hours a day u study ?

Commented by math solver last updated on 11/Feb/18

its not fixed , even sometimes it goes like i ′nt studied for a whole day .

its not fixed, even sometimes it goes

like i^{'} nt studied for a whole day .

Commented by 803jaideep@gmail.com last updated on 11/Feb/18

im general category... im from punjab patiala by d way

im general category \dots im from

punjab patiala by d way

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