A calculus problem by Anik Mandal

Let $\mathfrak{L}=\lim_{x\to\infty} (\sqrt{x^4+ax^3+3x^2+bx+2}- \sqrt{x^4+2x^3-cx^2+3x-d})$ Start by rationalising : $=\lim_{x\to\infty} \dfrac{((a-2)x^3+(c+3)x^2+(b-3)x+d+2)}{(\sqrt{x^4+ax^3+3x^2+bx+2}+ \sqrt{x^4+2x^3-cx^2+3x-d})}$ For a finite limit(=4) to exist , $a-2=0$ or $\large a=2$ . While the limit of the expression is: $\large \mathfrak L=\dfrac{c+3}{1+1}=\dfrac{c+3}{2}$ which is equal to 4, hence $\large c=5$ . $\huge \therefore a+c=\huge\color{#456461}{\boxed{\color{#D61F06}{\boxed{\color{#007fff}{\textbf{7}}}}}}$