Homework Help: Limit of a function

chiakimaron · May 13, 2008

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 · May 13, 2008

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

chiakimaron · May 13, 2008

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 · May 13, 2008

start with 0<a<1 then what happens?

chiakimaron · May 13, 2008

i really don't know

chiakimaron · May 13, 2008

limit x→∞〔 (a^x - 1 )/(a - 1) 〕^1/x

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

is it correct?

benorin · May 13, 2008

chiakimaron said: ↑

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 [tex]x\to\infty[/tex]

[tex]\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}[/tex]

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

malawi_glenn · May 13, 2008

chiakimaron said: ↑

limit x→∞〔 (a^x - 1 )/(a - 1) 〕^1/x

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

is it correct?

yes! :-)

And what if a>1 ?