Let f(x)=lim n→∞(x^2n(x+2)+x^2/(x^2n+1).

Evaluate the limit by cases. For $|x|<1$, $x^{2n}\to0$ so $f(x)=x^2$. For $|x|>1$, $x^{2n}\to\infty$ so $f(x)=x+2$. At $|x|=1$, $x^{2n}=1$ and $f(x)=\dfrac{x+2+x^2}{2}$, giving $f(1)=2$ and $f(-1)=1$. At $x=-1$: the left limit is $-1+2=1$, the right limit is $1$, and $f(-1)=1$, so $f$ is continuous and (A) is true. The left derivative there is $1$ while the right derivative is $2x=-2$, so (B) is false. At $x=1$: left limit $1$, right limit $3$, value $2$, a jump, so (C) is true. Finally $\int_0^2f=\int_0^1x^2dx+\int_1^2(x+2)dx=\tfrac13+\tfrac72=\tfrac{23}{6}$, not $\tfrac{11}{3}$, so (D) is false (numerical quadrature agrees).