Problem

Let $f(x,y)=x\sin(x^2+y)+x+y$ for $x,y\in(-\infty,\infty)$. Which of the following statement is true?

(a) $\displaystyle\lim_{(x,y)\to(0,0)}\frac{f(x,y)}{x(x^2+y)}=1$.

(b) $\displaystyle\int_0^a f(x,y)\,dy=-x\cos(x^2+a)+x\cos(x^2)+ax+\frac{a^2}{2}$ for $a>0$.

(c) $\dfrac{\partial}{\partial y}f(x,y)=2x^2\cos(x^2+y)+\sin(x^2+y)+1$.

(d) The tangent plane to the surface $z=f(x,y)$ at the point $(0,0,0)$ is $z=2x+y$.

(e) None of the above statements is true.