Please note that this problem was on a previous exam. Consider the equation

$\frac{dx}{dt}=(x^2-1)(x^4+1)(1+x)(2-x),x(0)=x_0$

$($a$)$ Find all critical points and determine their stability. If any critical point turns out to be semi$-$stable, indicate from which direction it is stable and which direction it is unstable.

$($b$)$ If $x_0=-0\mathrm{.}5$ determine $\underset{t}{lim}x(t).$ You should read this problem very carefully. You are not asked to find the exact solution of the problem $($and don't even try as the integral would be very hard$).$ You are just asked to find the limit$.$