Which expression is correct for using the sign $:=$ in the given problem?

Question

Imagine that I have this limit $$ \lim_{x\to a} f(x)=\frac{b^2-c^3}{8b^2-9c^2+a} ,$$ and I want to denote its RHS by another notation, say, $g$. Then, which notation is correct

$$ \lim_{x\to a} f(x)=\frac{b^2-c^3}{8b^2-9c^2+a}:=g ,$$ or $$ \lim_{x\to a} f(x)=g:=\frac{b^2-c^3}{8b^2-9c^2+a} ,$$

Answer 1

None of them are correct. I would write instead:

Setting $$g = \frac{b^2-c^3}{8b^2-9c^2+a},$$ one has $$\lim_{x \to a}f(x) = g.$$