TITLE: Evaluating the limit for a point on the curve
QUESTION [7 upvotes]: For a point $P(a,b)$ is a point lying on the curve satisfying $$2xy^2dx + 2x^2 y dy -
 \tan(x^2y^2) dx =0 $$ 
$\lim_{a\to -\infty}b = ? $
Options are:
  a) $ 0$ 
b) $-1 $
c) $1$ 
d) does not exist.

Attempt: 
If we observe carefully we get: 
$d(x^2 y^2) = \tan(x^2 y^2) dx$
$\implies \ln(c\sin x^2y^2) = x$
$\implies c \sin (x^2 y^2) = e^x$
Now clearly as $x \to -\infty ~ , e^x \to 0$,  so clearly $y \to 0$
But answer given is d. Please let me know my mistake.

REPLY [7 votes]: We have the following line:
$$c \sin (x^2 y^2) = e^x$$
The problem is that $\arcsin$ return only a single branch of $\sin$, not all of them:
Now, $c \sin (x^2 y^2) = e^{x}$ gives $$\sin (x^2 y^2) = \frac{e^{x}}c\implies x^2 y^2=\arcsin\left(\frac{e^{x}}c\right)\color{red}{+2k\pi}\\\implies y=\pm\frac1x\sqrt{\arcsin\left(\frac{e^{x}}c\right)\color{red}{+2k\pi}}$$So you have more then a single value for $y$ for any fixed $x$ and the answered are as big as we want so the limit won't get rid from them. So the limit is undefined