$\begin{aligned} & \lim _{x \rightarrow 0} \frac{\sqrt{1+2 x}-\sqrt{1-3 x}}{x} \\ & \end{aligned}$ Solve This Math

Solve This Math: $\begin{aligned} & \lim _{x \rightarrow 0} \frac{\sqrt{1+2 x}-\sqrt{1-3 x}}{x} \\ & \end{aligned}$$\begin{aligned} & \text { (Solve) } \lim _{x \rightarrow 0} \frac{\sqrt{1+2 x}-\sqrt{1-3 x}}{x} \\ & =\lim _{x \rightarrow 0} \frac{(\sqrt{1+2 x}-\sqrt{1-3 x})(\sqrt{1+2 x}+\sqrt{1-3 x})}{x(\sqrt{1+2 x}+\sqrt{1-3 x})} \\ & =\lim _{x \rightarrow 0} \frac{(\sqrt{1+2 x})^2-(\sqrt{1-3 x})^2}{x(\sqrt{1+2 x}+\sqrt{1-3 x})} \\ & =\lim _{x \rightarrow 0} \frac{1+2 x-1+3 x}{x(\sqrt{1+2 x}+\sqrt{1-3 x})} \\ & =\lim _{x \rightarrow 0} \frac{5 x}{x(\sqrt{1+2 x}+\sqrt{1-3 x})} \\ & =\lim _{x \rightarrow 0} \frac{5}{\sqrt{1+2 x}+\sqrt{1-3 x}} \\ & =\frac{5}{\sqrt{1+2.0}+\sqrt{1-3.0}} \\ & =\frac{5}{1+1} \\ & =\frac{5}{2} \text { (Ans.) } \\ & \end{aligned}$