1. **State the problem:** Find the limit $$\lim_{x \to x^+} \left( \frac{4x^2}{9} - x^2 \right)$$. 2. **Rewrite the expression:** $$\frac{4x^2}{9} - x^2 = \frac{4x^2}{9} - \frac{9x^2}{9} = \frac{4x^2 - 9x^2}{9} = \frac{-5x^2}{9}$$ 3. **Evaluate the limit:** Since the expression simplifies to $$\frac{-5x^2}{9}$$, and as $$x \to x^+$$ (approaching from the right), the value of $$x$$ approaches itself, so the limit is simply: $$\lim_{x \to x^+} \frac{-5x^2}{9} = \frac{-5x^2}{9}$$ (the expression is continuous in $$x$$). 4. **Final answer:** $$\boxed{\frac{-5x^2}{9}}$$ Note: The limit depends on the value of $$x$$ itself since the limit is as $$x \to x^+$$, which means approaching the value $$x$$ from the right side. This is a general expression, not a numeric limit.