Whats up? I have to freshman tier, brainlet questions for you.

1st) Are phasors a rigorously defined mathematical object? its very convenient to use them for circuit analysis and stuff, but the notion that a vector in the complex plane "spins" into a sine wave with an associated frequency just because seems very ass-pulled for me.

2nd) why is it that when you have a limit thats resulting on a infinity/infinity discontinuity you can divide numerador and denominator by the largest power term separately? As in:
$$\displaystyle \lim_{x \rightarrow \infty} \frac(x^2 + 3x + 5}{x^3 + 5x^2 + 4} = \displaystyle \lim_{x \rightarrow \infty} \frac{1/x + 3/x^2 + 5/x^3}{1 + 5/x + 4/x^3} = 0$$