Answer
$0$
Work Step by Step
Consider: $\lim\limits_{x \to \infty}f(x)=\lim\limits_{x \to \infty}\dfrac{2x^2 +3x}{x^3+x+1}$
Need to check that the limit has an indeterminate form.
Thus, $f(\infty)=\dfrac{\infty}{\infty}$
Now, apply L-Hospital's rule such as: $\lim\limits_{a \to b}f(x)=\lim\limits_{a \to b}\dfrac{g'(x)}{h'(x)}$
This implies:
$\lim\limits_{x \to \infty}\dfrac{4x+3}{3x^2+1}=\dfrac{\infty}{\infty}$
Again apply L-Hospital's rule.
$\lim\limits_{x \to \infty}\dfrac{4}{6x}=\dfrac{4}{\infty}=0$
