Limit of a function

  • #1
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
 

Answers and Replies

  • #2
malawi_glenn
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what have you tried? If you are not showing your effort, then you can't get any help
 
  • #3
let y=〔 (a^x - 1 )/(a - 1) 〕^1/x

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

then I can't do it
 
  • #4
malawi_glenn
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start with 0<a<1 then what happens?
 
Last edited:
  • #5
i really don't know
 
  • #6
limit x→∞〔 (a^x - 1 )/(a - 1) 〕^1/x

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

is it correct?
 
  • #7
benorin
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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.
 
  • #8
malawi_glenn
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limit x→∞〔 (a^x - 1 )/(a - 1) 〕^1/x

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

is it correct?
yes! :-)

And what if a>1 ?
 

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