Define f(x)=lim n→∞(x^2n x^2+x)/(x^2n+1) for real x.

For $|x|<1$, $x^{2n}\to0$ so $f(x)=x$. For $|x|>1$, $x^{2n}\to\infty$ and $f(x)=x^{2}$. At $x=\pm1$, $x^{2n}=1$, so $f(\pm1)=\frac{x^2+x}{2}$, giving $f(1)=1$ and $f(-1)=0$. (A) True: the left limit at $1$ is $1$, the right limit is $1^2=1$ and $f(1)=1$. (B) False: from the left $f'=1$, from the right $f'=2x=2$, so the one-sided derivatives differ and $f$ is not differentiable at $1$. (C) True: $f(-1)=\frac{1-1}{2}=0$ (note that $f$ is discontinuous there, since the left limit is $-1$ and the right limit is $1$). (D) False: $\int_0^2 f\,dx=\int_0^1x\,dx+\int_1^2x^2dx=\frac12+\frac73=\frac{17}{6}$, not $\frac{19}{6}$. Correct options: A and C.