Answer
$\infty$
Work Step by Step
L'Hospital's rule states that $\lim\limits_{x \to \infty} f(x)=\lim\limits_{x \to \infty} \dfrac{A'(x)}{B'(x)}$
Here, $\lim\limits_{x \to \infty} f(x)=\lim\limits_{x \to \infty} (\sqrt {x^3+1}-\sqrt x)$
or, $\lim\limits_{x \to \infty} (\sqrt {x^3+1}-\sqrt x)=\lim\limits_{x \to \infty} x (\dfrac{\sqrt {x^3+1}}{x}-\dfrac{\sqrt x}{x})$
or, $\lim\limits_{x \to \infty} (\sqrt {1+1/x^2}-\sqrt{\dfrac{1}{x}})=\lim\limits_{x \to \infty} (x)(1)=\infty$
