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Infinity is a concept that represents a value larger than any finite number. It is often depicted by the symbol \( \infty \), and while it's not a number in the traditional sense, it's used to describe quantities that grow without bound. In calculus, we frequently deal with limits approaching infinity, particularly when examining the behavior of functions as their input values become exceedingly large or small.

When we say that \( x \to \infty \), we're describing a situation where \( x \) increases without limit. In the context of limits, such as with the expression given in the problem \( \lim_{x \to \infty}(\sqrt{9x^2-x}-3x) \), the goal is to understand what happens to the function's value as \( x \) grows indefinitely.

It is important to remember that infinity itself can't be manipulated like an ordinary number; for instance, operations like \( \infty - \infty \) are undefined. When dealing with limits involving infinity, we often simplify expressions to more easily evaluate the limiting behavior as the variable grows without bound.

Square roots are mathematical operations that find the original number which was multiplied by itself to reach a given value. Notated as \( \sqrt{} \), the square root of a number \( a \) is a number \( b \) such that \( b^2 = a \).

In our given limit problem, we encounter a square root expression \( \sqrt{9x^2-x} \). To simplify solving the limit, the expression inside the square root can be rewritten after factoring out \( x^2 \), leading to \( \sqrt{x^2(9 - \frac{1}{x})} \).

This manipulation shows a critical property of square roots: \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \). The square root plays an important role in the simplification process—a common technique when dealing with limits, particularly when expressions inside square roots are involved. Additionally, when \( x \to \infty \), terms like \( \frac{1}{x} \) become negligible, which profoundly influences the behavior of the square root expression.

Simplifying expressions is a fundamental skill in algebra and calculus, aiming to reduce a complex expression into a simpler or more workable form, especially when evaluating limits. This involves factoring, combining like terms, or using mathematical identities to streamline the expression.

In the exercise, simplifying began by factoring out \( x^2 \) from under the radical of \( \sqrt{9x^2-x} \) to get \( x \sqrt{9-\frac{1}{x}} \). This simplification is crucial in revealing the internal structure of the limit expression, making it manageable to analyze as \( x \to \infty \).

The aim of simplifying here was to explicitly observe how the behavior of \( \frac{1}{x} \) becomes trivial as \( x \) grows very large, leaving us with more straightforward sub-expressions. It ultimately allowed us to review that \( \sqrt{9 - \frac{1}{x}} - 3 \to 0 \), leading to the entire expression becoming zero. Simplifying expressions often uncovers the hidden relationships and enables further reductions, making complex problems simpler and clearer.