Solve limfrac{1sqrt{1-x^2}}{3x}=limx(-1)/2

$$\lim \frac { 1 \sqrt { 1 - x ^ { 2 } } } { 3 x } = \lim \frac { x ( - 1 ) } { 2 }$$

Solve for l

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

Steps for Solving Linear Equation

Cancel out 1 in both numerator and denominator.

$$lIm(\frac{\sqrt{1-x^{2}}}{3x})=lIm(\frac{x\left(-1\right)}{2})$$

Subtract lIm(\frac{x\left(-1\right)}{2}) from both sides.

$$lIm(\frac{\sqrt{1-x^{2}}}{3x})-lIm(\frac{x\left(-1\right)}{2})=0$$

Reorder the terms.

$$-lIm(\frac{-x}{2})+Im(\frac{\sqrt{-x^{2}+1}}{3x})l=0$$

Combine all terms containing l.

$$\left(-Im(\frac{-x}{2})+Im(\frac{\sqrt{-x^{2}+1}}{3x})\right)l=0$$

The equation is in standard form.

$$\left(\frac{Im(x)}{2}+\frac{Re(\frac{1}{x})Im(\sqrt{1-x^{2}})}{3}+\frac{Im(\frac{1}{x})Re(\sqrt{1-x^{2}})}{3}\right)l=0$$

Divide 0 by \frac{1}{2}Im(x)+\frac{1}{3}Re(\sqrt{1-x^{2}})Im(x^{-1})+\frac{1}{3}Im(\sqrt{1-x^{2}})Re(x^{-1}).

$$l=0$$