Limit of Function x→∞: 0<a<1, a>1

[chiakimaron]

I have no idea of the following question, please help me

Find limit x→∞〔 (a^x - 1 )/(a - 1) 〕^1/x

for (1) 0<a<1
(2) a>1

 

[malawi_glenn]

what have you tried? If you are not showing your effort, then you can't get any help

 

[chiakimaron]

let y=〔 (a^x - 1 )/(a - 1) 〕^1/x

Iny=1/x In(a^x - 1 )/(a - 1)

then I can't do it

 

[malawi_glenn]

start with 0<a<1 then what happens?

 

[chiakimaron]

i really don't know

 

[chiakimaron]

limit x→∞〔 (a^x - 1 )/(a - 1) 〕^1/x

= [(-1)/(a-1)]^0 = 1

is it correct?

 

[benorin]

let y=〔 (a^x - 1 )/(a - 1) 〕^1/x

Iny=1/x In(a^x - 1 )/(a - 1)

then I can't do it

Keep going! Let $$x\to\infty$$

$$\lim_{x\to\infty}\frac{\ln\left(\frac{a^x-1}{a-1}\right)}{x}=\frac{\lim_{x\to\infty}\ln\left(\frac{a^x-1}{a-1}\right)}{\lim_{x\to\infty}x}$$

and remember ln(-1/(a-1)) is some constant.

 

[malawi_glenn]

limit x→∞〔 (a^x - 1 )/(a - 1) 〕^1/x

= [(-1)/(a-1)]^0 = 1

is it correct?

yes! :-)

And what if a>1 ?