Consider a continuous single variable real valued function $f(t)$ defined on some non empty finite interval $D$.
Say we wish to find the limit of $f(t)$ as $t$ approach some point $t_{d} \in D$ (here lets impose from above)
$$L = \lim_{t\rightarrow t_{d}^{+}} f(t) $$
Often we can apply a transformation on the limit to evaluate i.e. (log, exponential, trigonometric)-transforms. We then evaluate the transformed limit and finally then take the inverse transform to solve. For instance consider the following:
$$L = \lim_{t\rightarrow 0^{+}} t^t$$
As $t^t$ is continuous for $t > 0$ we can apply the log-transformation
$$\ln(L) = \lim_{t\rightarrow 0^{+}} \ln\left(t^t\right) = \lim_{t\rightarrow 0^{+}} t\ln\left(t\right) = 0$$
We then take the inverse log-transformation to find $L$
$$ L = e^{0} = 1 $$
My question is, under what conditions must we meet to use a given transform? Under what circumstances could I use say a Fourier transform?
Your problem seems to fall in the following general category:
You have a function $f(t)$ whose limit as $t$ tends to some $t_0 $ you cannot evaluate. However, you've also got some continuous bijection $g(t)$ such that you know the limit of $g(f(t))$ as $t$ tends to $t_0$. Then you can recover the initial limit $\lim_{t \to t_0}f(t) = \lim_{t \to t_0}g^{-1}(g(f(t))) = g^{-1}(\lim_{t \to t_0}g(f(t))) $, where the second equality follows by continuity of g.
In your example, $f(t)=t^t$ and $g(t)=\log t$ and $g^{-1}(t)=e^t$
So I guess you need continuity and bijectivity assumptions. Regarding your idea about using Fourier transforms, it is a little bit unclear to me what you have in mind, because a Fourier transform is a transformation of functions, a map from a space of functions to a space of functions, so I don't think it can be generally applied to problems with limits.