Chọn phương án D.

\(\begin{aligned}
\lim\limits_{x\to-1}\dfrac{x^5+1}{x^3+1}&=\lim\limits_{x\to-1}\dfrac{(x+1)\left(x^4-x^3+x^2-x+1\right)}{(x+1)\left(x^2-x+1\right)}\\
&=\lim\limits_{x\to-1}\dfrac{x^4-x^3+x^2-x+1}{x^2-x+1}\\
&=\dfrac{(-1)^4-(-1)^3+(-1)^2-(-1)+1}{(-1)^2-(-1)+1}\\
&=\dfrac{5}{3}.
\end{aligned}\)