To simplify the complex rational expression $\frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x}-\frac{1}{y}}$ by writing it as division, perform the following steps:

Step 1. Simplify the numerator and denominator. The common denominator for the separate expressions in the numerator and denominator is $xy$. Numerator: Multiply $\frac{1}{x}$ by $\frac{y}{y}$ and $\frac{1}{y}$ by $\frac{x}{x}$ to add them: $\frac{y}{xy} + \frac{x}{xy} = \frac{y + x}{xy}$. Denominator: Multiply $\frac{1}{x}$ by $\frac{y}{y}$ and $\frac{1}{y}$ by $\frac{x}{x}$ to subtract them: $\frac{y}{xy} - \frac{x}{xy} = \frac{y - x}{xy}$. The complex fraction becomes $\frac{\frac{y + x}{xy}}{\frac{y - x}{xy}}$.

Step 2. Rewrite as division. Translate the main fraction bar into division: $\frac{y + x}{xy} \div \frac{y - x}{xy}$.

Step 3. Multiply by the reciprocal and simplify. Change the division operator to multiplication by using the reciprocal: $\frac{y + x}{xy} \cdot \frac{xy}{y - x}$. Divide out the common polynomial factor of $xy$ from the numerator and denominator. The final simplified expression is $\frac{y + x}{y - x}$.