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A horizontal asymptote is a horizontal line that a function approaches as the input, usually denoted by \( x \), becomes very large or very small. In simpler terms, as you move far to the right or far to the left along the x-axis on a graph, the value of the function gets closer and closer to the horizontal asymptote.
In the context of the function \( h(x) = \frac{-5 + \frac{7}{x}}{3 - \frac{1}{x^2}} \), we've calculated that as \( x \to \infty \) and \( x \to -\infty \), \( h(x) \) approaches \( \frac{-5}{3} \).
This means the horizontal asymptote of the function is the line \( y = \frac{-5}{3} \).

• The points \( \frac{7}{x} \) and \( \frac{1}{x^2} \) weaken as \( x \) increases or decreases extensively because they are divided by large numbers.
• The main terms \( -5 \) and \( 3 \) remained constant, leading the value towards \( \frac{-5}{3} \).

This helps to tell us how the function behaves for values that are way out on either side of the graph.

Infinity is not a number, but a concept. It's like saying something is immeasurably large. Understanding infinity in mathematics helps us determine what happens to a value when it grows without bounds or shrinks beyond measurable limits.
When we say \( x \to \infty \), it means that \( x \) increases towards larger and larger positive values.
For \( x \to -\infty \), \( x \) is instead moving towards larger negative values.
In our exercise, when calculating the limit of \( h(x) \) as \( x \to \infty \) or \( x \to -\infty \), we simplify the formula by neglecting terms with \( x \) in the denominator. This is because their impact diminishes to zero as \( x \) approaches infinity.

• As \( \frac{7}{x} \to 0 \), the fraction diminishes since 7 is divided by an enormous number.
• Similarly, \( \frac{1}{x^2} \to 0 \) occurs even faster as any large number squared becomes even larger.

Thus, understanding infinity helps identify how certain terms in a function cease to contribute significantly as \( x \) moves towards extremely large or small values.

Rational functions are expressions that involve fractions, with both the numerator and the denominator being polynomials.
The function \( h(x) = \frac{-5 + \frac{7}{x}}{3 - \frac{1}{x^2}} \) is an example of a rational function. Here, we're dealing with terms \( -5 + \frac{7}{x} \) in the numerator and \( 3 - \frac{1}{x^2} \) in the denominator.
In rational functions, a key feature is their behavior at values of \( x \) where the denominator may become zero, unless specified otherwise, would result in undefined values.
Understanding how to find limits for rational functions involves:

• Simplifying by ignoring parts of the formula that become negligible at extreme values of \( x \).
• Recognizing dominant terms that dictate the function's behavior at infinity or negative infinity.

In our case, all polynomial terms divided by \( x \) or \( x^2 \) diminish, letting the prevailing constants dictate the approach to the horizontal asymptote. This simplifies calculation and helps readily understand how rational functions change over very large or very small inputs.