Solve N=lim(sqrt{4n^2+1}-sqrt[3]{8n^3+n})

$$N = \lim ( \sqrt { 4 n ^ { 2 } + 1 } - \sqrt[ 3 ] { 8 n ^ { 3 } + n } )$$

Solve for l

$$\left\{\begin{matrix}l=\frac{N}{Im(\sqrt{4n^{2}+1})-Im(\sqrt[3]{8n^{3}+n})}\text{, }&Im(\sqrt{4n^{2}+1})-Im(\sqrt[3]{8n^{3}+n})\neq 0\\l\in \mathrm{C}\text{, }&N=0\text{ and }Im(\sqrt{4n^{2}+1})-Im(\sqrt[3]{8n^{3}+n})=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$lIm(\sqrt{4n^{2}+1}-\sqrt[3]{8n^{3}+n})=N$$

The equation is in standard form.

$$\left(Im(\sqrt{4n^{2}+1})-Im(\sqrt[3]{8n^{3}+n})\right)l=N$$

Divide both sides by Im(\sqrt{4n^{2}+1})-Im(\sqrt[3]{8n^{3}+n}).

$$\frac{\left(Im(\sqrt{4n^{2}+1})-Im(\sqrt[3]{8n^{3}+n})\right)l}{Im(\sqrt{4n^{2}+1})-Im(\sqrt[3]{8n^{3}+n})}=\frac{N}{Im(\sqrt{4n^{2}+1})-Im(\sqrt[3]{8n^{3}+n})}$$

Dividing by Im(\sqrt{4n^{2}+1})-Im(\sqrt[3]{8n^{3}+n}) undoes the multiplication by Im(\sqrt{4n^{2}+1})-Im(\sqrt[3]{8n^{3}+n}).

$$l=\frac{N}{Im(\sqrt{4n^{2}+1})-Im(\sqrt[3]{8n^{3}+n})}$$

Solve for N

$$N=l\left(-Im(\sqrt[3]{n\left(8n^{2}+1\right)})+Im(\sqrt{4n^{2}+1})\right)$$