Is $\dfrac{0}{0}$ indeterminate with respect to limit only like $\lim_{x\to 0}\dfrac{\sin x}{x}$ or it is indeterminate in general like $\log_1 1$?
I asked this because I encountered indeterminates only in limits, we don't talk about in other areas.
Like $\log_1 1$ can be any real number.
You should probably be more explicit in phrasing your Question.
Certainly "$\frac{0}{0}$" is shorthand for the indeterminacy of limits in the first sense, but whether the expressions of a more general form are indeterminate will depend on defining them in a limiting sense or otherwise.
I don't see any real objection to saying that $1^0 = 1$, but then again $1$ to any power can be $1$. In that respect we would rather say that $log_1 1$ is not a well-defined operation (rather than call it indeterminate). In an application where a well-defined value is indicated, one would have to backtrack to discover where critical information has been lost or forgotten.
The point of calling something "indeterminate" is that further information is needed to arrive at a value. Thus it is not enough information to determine a value to know the numerator goes to a limit of zero and the denominator goes to a limit of zero. The limit of such ratios might not exist, or it might exist but be any value at all, without further information.