Evaluate lim x→∞: Evaluating Limit with x

[mathlike]

1. Homework Statement [/b]

Evaluate lim x goes to ∞ positive
((x^-1)*(b^x-1)/(b-1))^(1/x)

2. Homework Equations

3. The Attempt at a Solution [/b]
I try to take log but it does not seem to work

 

[DivisionByZro]

Is this the problem?

<br /> \lim_{x\to\infty}\frac{(b^{x}-1)}{x(b-1)}^{\frac{1}{x}}<br />

 

[mathlike]

Yes. I tried taking log but to no avail

 

[alanlu]

http://en.m.wikipedia.org/wiki/Squeeze_theorem may be useful for this problem. What is the numerator bounded by?

 

[mathlike]

still cannot solve it. Need help guys! Appreciated.

 

[alanlu]

To clarify, is it this?
<br /> \lim_{x\to\infty}\left(\frac{b^{x}-1}{x(b-1)}\right)^{\frac{1}{x}}<br />
Or is it this?
<br /> \lim_{x\to\infty}\frac{(b^{x}-1)^{\frac{1}{x}}}{x(b-1)}<br />

For the latter, note that for all x > 1, b > 1,
$$0 \leq (b^{x}-1)^{\frac{1}{x}} \leq b$$
For the former, I am not immediately sure.

 

[Dick]

Judging by you initial parentheses I think you actually mean. \lim_{x\to\infty} \big( \frac{b^{x}-1}{x(b-1)} \big) ^{\frac{1}{x}}.

And can you show why taking the log isn't working. Or something?

 

[mathlike]

I simplified it down to calculate
limx->infinity (b^x-1)/x
...can help me to proceed?

 

[Dick]

I simplified it down to calculate
limx->infinity (b^x-1)/x
...can help me to proceed?

I don't see how you got that. You need to show us.