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Understanding the limit of a sequence is key to determining whether a sequence converges or diverges. A sequence is said to converge if it approaches a specific number as it progresses to infinity. Conversely, it diverges if it doesn't settle on a particular number.

For any given sequence \( a_n \), the limit \( \lim_{n \to \infty} a_n \) describes the value that the sequence approaches as \( n \) goes to infinity. If a sequence converges, the limit gives its long-term behavior. In the problem, simplifying \( a_n = n - 5 \) shows the sequence increases indefinitely as \( n \to \infty \), thus diverging.

Factorization is a technique used to simplify expressions, especially in sequences and polynomial expressions. It involves breaking down an expression into its constituent factors.

In our exercise, the expression \( n^2 - 25 \) in the numerator is a classic example of a difference of squares. The 'Difference of Squares' can be recognized by the formula \( a^2 - b^2 = (a-b)(a+b) \), where \( n^2 - 25 \) is factored into \((n-5)(n+5)\). This step is crucial for simplifying sequences further since the common factors can be canceled out, leading to a much more straightforward form of the sequence that simplifies analysis and limit computation.

The 'Difference of Squares' is a fundamental algebraic identity, often used in simplifications. It takes the form \( a^2 - b^2 = (a-b)(a+b) \) and is particularly useful in sequences and function simplifications.

In this problem, the difference \( n^2 - 25 \) is neatly expressed as \((n-5)(n+5)\), thanks to this identity. Recognizing this pattern allows for efficient simplification because these factors can often be canceled with similar terms in the denominator. Here, cancelation of \( n+5 \) with the denominator simplifies the sequence considerably, making it easier to identify its limit and behavior.

Sequence simplification involves reducing a sequence to its most basic form, allowing for easier analysis. This may include factoring, canceling common terms, and reducing expressions.

From the original sequence \( a_n = \frac{n^2-25}{n+5} \), simplification is achieved by factoring the numerator using the 'Difference of Squares' formula. When the expression \( n^2 - 25 \) is expanded to \((n-5)(n+5)\), and \( n+5 \) cancels with the denominator, we are left with \( a_n = n-5 \). This simplified form of \( a_n \) shows immediately that as \( n \to \infty \), \( a_n \) diverges to infinity. This simplification process is fundamental in identifying behavior trends in sequences, helping us determine convergence or divergence efficiently.