1. The problem states that $x = 0.999999\ldots$ and asks about the equality $0.999999\ldots = \frac{9}{9}$ and whether $\frac{9}{9} = 1$.\n\n2. First, note that $\frac{9}{9} = 1$ because any nonzero number divided by itself equals 1.\n\n3. Now, let's analyze the decimal $0.999999\ldots$. This is a repeating decimal where the digit 9 repeats infinitely.\n\n4. A well-known fact in mathematics is that $0.999999\ldots$ (with infinite 9s) is exactly equal to 1.\n\n5. To prove this, consider the variable $x = 0.999999\ldots$. Multiply both sides by 10:\n$$10x = 9.999999\ldots$$\n\n6. Subtract the original $x$ from this equation:\n$$10x - x = 9.999999\ldots - 0.999999\ldots$$\n$$9x = 9$$\n\n7. Divide both sides by 9:\n$$x = 1$$\n\n8. Since $x$ was defined as $0.999999\ldots$, this shows $0.999999\ldots = 1$.\n\n9. Therefore, the statement $0.999999\ldots = \frac{9}{9} = 1$ is true.\n\nFinal answer: $0.999999\ldots = 1$.