I'm computing the lower bound for the Hausdorff dimension of a Cantor-like set; I've reduced it to computing $\lim_{k\rightarrow \infty, \delta \rightarrow 0^+}\frac{1}{(1+\delta)^{k(p-1)}}$, where $p$ is some prime. This limit depends on the relationship between $\delta, k$, otherwise it is undefined. (http://math.stackexchange.com/questions/204602/computing-a-double-limit)
I arrived to this limit by playing with an expression (for which no relation between $\delta, k$ was specified, but we eventually want to take the limit as $\delta$ tends to 0) $(*) \lim_{k\rightarrow \infty}\frac{\log(m_p*...*m_{kp})}{\log(n_{k+1})}=\lim_{k\rightarrow \infty}\frac{\log(n_0^{1+\delta}*...*n_0^{(1+\delta)^p})}{\log(n_0^{(1+\delta)^kp})}=\frac{(1+\delta)+...+(1+\delta)^p}{(1+\delta)^{kp}}$ I simplified this limit to get that it is $\ge \lim \frac{1}{(1+\delta)^{k(p-1)}}$
Note all the $\delta$s that appear are the same $\delta$. In the process of arriving at this limit, I never assumed there was a relationship between $\delta$ and $k$. Since I want the value of this limit to be a positive number less than $1$, I want to pick a specific relationship between $\delta$ and $k$ that will allow this to happen- which I am allowed to do. However, once I define a relationship $\delta_k=f(k)$, will this violate the steps I used to arrive to my limit expression if it involved $\delta, k$?
My instinct says yes because once I define $\delta_k=f(k)$, then each $\delta$ in the last equation of $(*)$ will be different, since it is coming from a different index $k$. Therefore, my steps need to be reconsidered.
On the other hand, all that was known originally that $\delta \rightarrow 0^+$, so why should it matter which path I choose for $\delta$ to approach zero? And therefore I should be able to do this without adjusting the steps used to get my limit lower bound expression.
Do I have to readjust my calculation and place a $\delta_k$ wherever there is a $\delta$ once I specify this relationship?