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Horizontal asymptotes are a concept in calculus that helps us understand the behavior of a function as the input variable — usually denoted as \( x \) — moves towards large positive or negative values. In simpler terms, they indicate the direction in which the function values are heading when \( x \) becomes either very large or very small.
Consider the function \( f(x) = \frac{2}{x} - 3 \). Both components play crucial roles.

• The \( \frac{2}{x} \) part decreases towards zero as \( x \) increases or decreases.
• The constant \(-3\) remains unchanged.

Therefore, as \( x \) becomes very large in magnitude, the function value tends towards \(-3\). This constant value is where the horizontal asymptote lies, at \( y = -3 \).
This means if you were to graph this function, no matter how far you zoomed out, the graph would always approach the line \( y = -3 \), but never quite touch it. The asymptote guides the end behavior of the function.

Infinity in mathematics represents an unbounded limit. It's not a number, but rather a concept that describes something without any end. When we consider limits, saying \( x \rightarrow \infty \) suggests that the variable \( x \) grows larger and larger without confine.
For example, in the function \( f(x) = \frac{2}{x} - 3 \), evaluating the limit as \( x \rightarrow \infty \) means observing what happens to the function values as \( x \) increases:

• The term \( \frac{2}{x} \) approaches zero as \( x \) gets infinitely large.
• Subtracting 3 from zero, the output approaches \(-3\).

This tells us that the function effectively "levels off" and approaches the horizontal asymptote as \( x \) becomes infinitely large. Each increment in \( x \) makes the fraction smaller, demonstrating the concept of infinity in action.

Negative infinity is similar to the concept of infinity but in the opposite direction. When \( x \rightarrow -\infty \), it indicates that \( x \) is growing larger in the negative direction. This concept helps us understand how functions behave as their inputs decrease without bound.
For \( f(x) = \frac{2}{x} - 3 \), consider the limit as \( x \rightarrow -\infty \):

• The term \( \frac{2}{x} \) approaches zero as \( x \) becomes more negatively large.
• Thus, our function tends towards the same horizontal asymptote \( y = -3 \) just as it does when \( x \) approaches positive infinity.

Effectively, the function behaves similarly in both "endless" directions, and the negative infinity perspective confirms that the graph approaches the asymptote \( y = -3 \) from below.