$\lim\left({\sqrt[3]{n^3+an+1}-\sqrt{n^2+an}}\right)$
$= \lim\left({\sqrt[3]{n^3+an+1}-n+n-\sqrt{n^2+an}}\right)$
$=\lim\left({ \dfrac{an+1}{\left({\sqrt[3]{n^3+an+1}}\right)^2+n.\sqrt[3]{n^3+an+1}+n^2} -\dfrac{an}{n+\sqrt{n^2+an}}}\right)$
$=\lim \left({{ \dfrac{\frac{a}{n}+\frac{1}{n^2}}{\left({\sqrt[3]{1+\frac{a}{n^2}+\frac{1}{n^3}}}\right)^2+\sqrt[3]{1+\frac{a}{n^2}+\frac{1}{n^3}}+1}} -\dfrac{a}{1+\sqrt{1+\frac{a}{n}}}}\right)$
$=\dfrac{0}{1+1+1}-\dfrac{a}{1+1}$
$=\dfrac{-a}{2}=-1$
$\to a=2$
