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What does the horizontal asymptote of a rational function represent?

In the function \[f(x) = \frac{x + 1}{x – 5},\] what happens at \( x = 5 \)?

The graph of \( f(x) = \frac{1}{x^2} \) differs from \( f(x) = \frac{1}{x} \) in which way?

Add the following rational expressions: \[\frac{4}{x – 1} + \frac{x}{x + 1}\]

Which polynomial could represent the graph? The x-axis spans from below zero to 5, and the y-axis spans from just below 0 to above 20. Both axes have a scale of 5 with grid lines in increments of 1. The green quartic polynomial function has two local minima and one local maximum. It starts from positive infinity in the second quadrant, decreases to a local minimum at (negative 1, 0), rises to a local maximum at (1, 16), and then falls to another local minimum at (3, 0) before increasing towards positive infinity in the first quadrant.  

Subtract the following rational expressions: \[\frac{x^2}{x + 1} – \frac{2x}{x – 1}\]

What is the multiplicity of the zero at \( x = -1 \)? The x-axis spans from below 0 to above 1, and the y-axis spans from below negative 5 to above 0. The x-axis has a scale of 1 in increments of 0.2 and the y-axis has a scale of 5 in increments of 1. The purple function has multiple turning points: it starts from negative infinity in the third quadrant, rises to a local maximum near the origin (0, 0), dips into a local minimum slightly below the point (0.5, negative 1) in the fourth quadrant, increases again to another local maximum around (1, 0), and then falls steeply towards negative infinity. The curve is symmetrical around vertical line passing through the local minimum.

Which factors of the polynomial shown in the graph have a multiplicity greater than one?  The x-axis spans from slightly below negative 2 to above 2, and the y-axis spans from below 0 to above 20. The x-axis has a scale of 2 in increments of 0.5 and the y-axis has a scale of 10 in increments of 2. The orange function exhibits multiple turning points: it starts from positive infinity, descends into a local minimum around (negative 1, 0), rises to a local maximum near (0.5, 15), dips again into another local minimum at (2, 0), and then increases steeply towards positive infinity. The curve passes through the y-axis at (0, 12). It is symmetrical around the vertical line passing through the local maximum.

Which of the following statements is true about the graph of \( f(x) = \frac{1}{x} \)?

Simplify the product: \[\frac{x^2 – 9}{x + 3} \cdot \frac{x + 2}{x – 3}\]