The vector field is $\vec F(x,y)=P(x,y)\hat i+Q(x,y)\hat j$. While:
$P(x,y)=\frac{4yx^7}{\sqrt{x^8+y^6}}$ whenever $(x,y)\ne0$ and otherwise equal $0$.
$Q(x,y)=\frac{3y^6}{\sqrt{x^8+y^6}}+\sqrt{x^8+y^6}$ whenever $(x,y)\ne0$ and otherwise equal $0$.
First of all, I know that if $Q_x\ne P_y$ the field isn't conservative, I went up and checked that and found that:
$P_y=Q_x=4x^7(\sqrt{x^8+y^6}-\frac{3y^6}{\sqrt{x^8+y^6}})$.
And now I'm a little confused on what to do, I know that if $P,Q$ have a continuous partial derivatives in a simply connected space. and $P_y=Q_x$
in every point in that space. then $\vec F=Pi+Qj$ is a conserative field in that space.
Now in order to prove continuity of $P_y,Q_x$ or even $P,Q$ themselves, I need to show that $\lim_{(x,y)}\to0$. But I'm struggling to calculate the limit.
Would appreciate any help and feedback, thanks in advance!
$P_x|_{(x,y)=(0,0)}=\operatorname{lim}_{h\rightarrow 0}\frac{P(h,0)-P(0,0)}{h}=0$. Similarly you can find $Q_y|_{(x,y)=(0,0)}=0$. Now it is easy to easy the term $4x^7(\sqrt{x^8+y^6})\rightarrow 0 $ as $(x,y)\rightarrow 0$. The other term is $\frac{12x^7y^6}{\sqrt{x^8+y^6}}$. Now $y^6\leq (x^8+y^6)$. Hence $|\frac{12x^7y^6}{\sqrt{x^8+y^6}}|\leq \frac{12|x|^7(x^8+y^6)}{\sqrt{x^8+y^6}}=12|x|^7\sqrt{x^8+y^6}\rightarrow0$ as $(x,y)\rightarrow 0$. This shows the continuity of $Q_y,P_x$ at $0$. The continuity at all other points are trivial. As $\mathbb{R}^2$ is obviously simply connected, you are done.
EDIT: I have simplified the calculation a little.