Understanding Horizontal Asymptotes and Points of Discontinuity in Functions.

Horizontal Asymptote at y=0 and 3 points of discontinuity

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Horizontal asymptote at y=0:
A horizontal asymptote is a straight line that the graph of a function approaches but never touches as x approaches positive or negative infinity. If a function has a horizontal asymptote at y=0, it means that the function approaches the x-axis but never touches it as x gets very large or very small.

Here’s an example of a function with a horizontal asymptote at y=0:
f(x) = (2x^2 + 3x – 1) / (4x^2 + 5x – 2)
To find the horizontal asymptote of this function, we first need to determine the highest degree of x in both the numerator and denominator. In this case, it’s x^2. So, as x approaches infinity or negative infinity, the terms with x^2 in the numerator and denominator become dominant. Therefore, we can simplify this function as follows:

f(x) ≈ (2x^2 / 4x^2) = 1/2

As x gets very large or very small, f(x) approaches 1/2, which is a horizontal line at y=0. Therefore, y=0 is the horizontal asymptote of this function.

Points of discontinuity:
A point of discontinuity is a point on the graph of a function where the value of the function is undefined or where the function jumps or changes abruptly. Here are three points of discontinuity for a function:

1. Vertical asymptote: A vertical asymptote occurs when the denominator of a fraction becomes zero. For example, the function g(x) = 1 / (x^2 – 4) has a vertical asymptote at x=2 and x=-2, because the denominator becomes zero at those points.

2. Hole: A hole (or removable discontinuity) occurs when a factor of the numerator and denominator cancel out. For example, the function h(x) = (x^2 – 4) / (x – 2) has a hole at x=2, because the factor (x-2) appears in both the numerator and denominator.

3. Jump: A jump discontinuity occurs when the function jumps or changes abruptly at a certain point. For example, the function j(x) = |x| has a jump discontinuity at x=0, because the value of the function changes from -x to x as x approaches 0 from negative and positive values, respectively.

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