Question

If $\displaystyle\lim _{x \rightarrow \infty}\left(\frac{x^{3}+1}{x^{2}+1}-(a x +b)\right)=2$, then find the value of $10 a+3 b$.

Answer 1

$\displaystyle\lim _{x \rightarrow \infty}\left(\frac{x^{3}+1}{x^{2}+1}-a x-b\right)$
$=\displaystyle\lim _{x \rightarrow \infty}\left(\frac{x^{3}(1-a)-b x^{2}-a x+1-b}{x^{2}+1}\right)$
Given that the limit is finite
$\Rightarrow 1-a=0$ and $b=2$
$\Rightarrow a=1$ and $b=-2$
$\Rightarrow 10 a+3 b=10-6=4$