Solve D=limfrac{sqrt{n^2+1}-sqrt[3]{3n^3+2}}{sqrt[4]{2n^4+n+2}-n}

$$D = \lim \frac { \sqrt { n ^ { 2 } + 1 } - \sqrt[ 3 ] { 3 n ^ { 3 } + 2 } } { \sqrt[ 4 ] { 2 n ^ { 4 } + n + 2 } - n }$$

Solve for l

$$\left\{\begin{matrix}l=\frac{D}{Re(\frac{1}{\sqrt[4]{2n^{4}+n+2}-n})Im(\sqrt{n^{2}+1})+Im(\frac{1}{\sqrt[4]{2n^{4}+n+2}-n})Re(\sqrt{n^{2}+1})-Re(\frac{1}{\sqrt[4]{2n^{4}+n+2}-n})Im(\sqrt[3]{3n^{3}+2})-Im(\frac{1}{\sqrt[4]{2n^{4}+n+2}-n})Re(\sqrt[3]{3n^{3}+2})}\text{, }&Re(\frac{1}{\sqrt[4]{2n^{4}+n+2}-n})\left(Im(\sqrt{n^{2}+1})-Im(\sqrt[3]{3n^{3}+2})\right)+Im(\frac{1}{\sqrt[4]{2n^{4}+n+2}-n})\left(Re(\sqrt{n^{2}+1})-Re(\sqrt[3]{3n^{3}+2})\right)\neq 0\text{ and }\sqrt[4]{2n^{4}+n+2}-n\neq 0\\l\in \mathrm{C}\text{, }&D=0\text{ and }Re(\frac{1}{\sqrt[4]{2n^{4}+n+2}-n})\left(Im(\sqrt{n^{2}+1})-Im(\sqrt[3]{3n^{3}+2})\right)+Im(\frac{1}{\sqrt[4]{2n^{4}+n+2}-n})\left(Re(\sqrt{n^{2}+1})-Re(\sqrt[3]{3n^{3}+2})\right)=0\text{ and }\sqrt[4]{2n^{4}+n+2}-n\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$lIm(\frac{\sqrt{n^{2}+1}-\sqrt[3]{3n^{3}+2}}{\sqrt[4]{2n^{4}+n+2}-n})=D$$

The equation is in standard form.

$$\left(Re(\frac{1}{\sqrt[4]{2n^{4}+n+2}-n})\left(Im(\sqrt{n^{2}+1})-Im(\sqrt[3]{3n^{3}+2})\right)+Im(\frac{1}{\sqrt[4]{2n^{4}+n+2}-n})\left(Re(\sqrt{n^{2}+1})-Re(\sqrt[3]{3n^{3}+2})\right)\right)l=D$$

Divide both sides by \left(Re(\sqrt{n^{2}+1})-Re(\sqrt[3]{3n^{3}+2})\right)Im(\left(\sqrt[4]{2n^{4}+n+2}-n\right)^{-1})+\left(Im(\sqrt{n^{2}+1})-Im(\sqrt[3]{3n^{3}+2})\right)Re(\left(\sqrt[4]{2n^{4}+n+2}-n\right)^{-1}).

$$\frac{\left(Re(\frac{1}{\sqrt[4]{2n^{4}+n+2}-n})\left(Im(\sqrt{n^{2}+1})-Im(\sqrt[3]{3n^{3}+2})\right)+Im(\frac{1}{\sqrt[4]{2n^{4}+n+2}-n})\left(Re(\sqrt{n^{2}+1})-Re(\sqrt[3]{3n^{3}+2})\right)\right)l}{Re(\frac{1}{\sqrt[4]{2n^{4}+n+2}-n})\left(Im(\sqrt{n^{2}+1})-Im(\sqrt[3]{3n^{3}+2})\right)+Im(\frac{1}{\sqrt[4]{2n^{4}+n+2}-n})\left(Re(\sqrt{n^{2}+1})-Re(\sqrt[3]{3n^{3}+2})\right)}=\frac{D}{Re(\frac{1}{\sqrt[4]{2n^{4}+n+2}-n})\left(Im(\sqrt{n^{2}+1})-Im(\sqrt[3]{3n^{3}+2})\right)+Im(\frac{1}{\sqrt[4]{2n^{4}+n+2}-n})\left(Re(\sqrt{n^{2}+1})-Re(\sqrt[3]{3n^{3}+2})\right)}$$

Dividing by \left(Re(\sqrt{n^{2}+1})-Re(\sqrt[3]{3n^{3}+2})\right)Im(\left(\sqrt[4]{2n^{4}+n+2}-n\right)^{-1})+\left(Im(\sqrt{n^{2}+1})-Im(\sqrt[3]{3n^{3}+2})\right)Re(\left(\sqrt[4]{2n^{4}+n+2}-n\right)^{-1}) undoes the multiplication by \left(Re(\sqrt{n^{2}+1})-Re(\sqrt[3]{3n^{3}+2})\right)Im(\left(\sqrt[4]{2n^{4}+n+2}-n\right)^{-1})+\left(Im(\sqrt{n^{2}+1})-Im(\sqrt[3]{3n^{3}+2})\right)Re(\left(\sqrt[4]{2n^{4}+n+2}-n\right)^{-1}).

$$l=\frac{D}{Re(\frac{1}{\sqrt[4]{2n^{4}+n+2}-n})\left(Im(\sqrt{n^{2}+1})-Im(\sqrt[3]{3n^{3}+2})\right)+Im(\frac{1}{\sqrt[4]{2n^{4}+n+2}-n})\left(Re(\sqrt{n^{2}+1})-Re(\sqrt[3]{3n^{3}+2})\right)}$$

Divide D by \left(Re(\sqrt{n^{2}+1})-Re(\sqrt[3]{3n^{3}+2})\right)Im(\left(\sqrt[4]{2n^{4}+n+2}-n\right)^{-1})+\left(Im(\sqrt{n^{2}+1})-Im(\sqrt[3]{3n^{3}+2})\right)Re(\left(\sqrt[4]{2n^{4}+n+2}-n\right)^{-1}).

$$l=\frac{D}{Re(\frac{1}{\sqrt[4]{2n^{4}+n+2}-n})Im(\sqrt{n^{2}+1})+Im(\frac{1}{\sqrt[4]{2n^{4}+n+2}-n})Re(\sqrt{n^{2}+1})-Re(\frac{1}{\sqrt[4]{2n^{4}+n+2}-n})Im(\sqrt[3]{3n^{3}+2})-Im(\frac{1}{\sqrt[4]{2n^{4}+n+2}-n})Re(\sqrt[3]{3n^{3}+2})}$$

Solve for D

$$D=\left(Re(\frac{1}{\sqrt[4]{2n^{4}+n+2}-n})\left(Im(\sqrt{n^{2}+1})-Im(\sqrt[3]{3n^{3}+2})\right)+Im(\frac{1}{\sqrt[4]{2n^{4}+n+2}-n})\left(Re(\sqrt{n^{2}+1})-Re(\sqrt[3]{3n^{3}+2})\right)\right)l$$

$$\sqrt[4]{2n^{4}+n+2}-n\neq 0$$