91Ó°ÊÓ

The signum function, often represented as \( \text{sgn}(x) \), plays a key role in understanding how we interpret the sign of numbers. It tells us whether a number is positive, negative, or zero.
For any given real number \( x \):

• If \( x > 0 \), \( \text{sgn}(x) = 1 \)
• If \( x = 0 \), \( \text{sgn}(x) = 0 \)
• If \( x < 0 \), \( \text{sgn}(x) = -1 \)

In our exercise, as \( x \) approaches infinity, which means \( x \) is getting larger indefinitely, the element of \( x \) being positive is central. When \( x \) becomes very large and positive, \( \text{sgn}(x) \) will consistently equal 1. Thus, the signum function simplifies complex scenarios by giving a straightforward indicator of the sign of a number.

An infinite limit is a concept in calculus where the variable within the limit approaches either positive infinity \( \infty \) or negative infinity \( -\infty \). This describes the behavior of function values as one moves further and further along the number line without bound.
In mathematical notation, \( \lim_{x \rightarrow \infty} f(x) \) describes the value that \( f(x) \) approaches as \( x \) becomes extremely large.
This is crucial because it helps us to determine the behavior of functions over extensive ranges:

• If \( \lim_{x \rightarrow \infty} f(x) = L \), where \( L \) is a constant, it tells us that \( f(x) \) stabilizes or approaches a horizontal asymptote at \( y = L \).
• If \( \lim_{x \rightarrow \infty} f(x) = \pm \infty \), it indicates the function grows without bound.

In our exercise, \( \lim_{x \rightarrow \infty} (\text{sgn}(x))^x \) involves \( \text{sgn}(x) \) becoming 1, and thus the expression behaves like \( 1^x \) which equals 1, showing stability.

Evaluating limits involves understanding the behavior of expressions as variables approach a certain value, potentially resulting in simpler expressions. The most common techniques to evaluate limits include:

• Substitution: Directly replace the variable with a given value, if no indeterminate form arises.
• Factoring: Break down complex expressions to cancel terms and simplify the limit evaluation.
• Hopital's Rule: Applied when encountering indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).

In our example, the limit \( \lim_{x\rightarrow\infty}(\text{sgn}(x))^x \), is simplified by recognizing that for \( x \rightarrow \infty \):

• The \( \text{sgn}(x) \) becomes 1.
• Reducing the complexity of the expression to \( 1^x \).
• Since any number to the power of \( x \) remains the same constant when \( x \) is infinity, the limit remains \( 1 \).

Hence, proper identification of terms and their behavior leads to an effective and concise solution of the problem.