Understanding Horizontal and Vertical Asymptotes: Explained with Examples and Rules

Horizontal Asymptote at y=3 and Vertical Asymptote at x = 4

To understand the concept of horizontal and vertical asymptotes, let’s analyze each of them separately

To understand the concept of horizontal and vertical asymptotes, let’s analyze each of them separately.

1. Horizontal asymptote at y = 3:
A horizontal asymptote is a line that a function approaches as x approaches positive or negative infinity. In this case, we have a horizontal asymptote at y = 3. This means that as x approaches positive or negative infinity, the function will get closer and closer to the line y = 3.

To determine the behavior of the function as x approaches infinity, we can look at the highest power of x in the function. Let’s assume the function is f(x).

a) If the highest power of x in f(x) is a positive even number, then as x approaches infinity, the function will approach the horizontal asymptote from both above and below.
b) If the highest power of x in f(x) is a positive odd number, then as x approaches infinity, the function will approach the horizontal asymptote, but it will come from either above or below depending on the sign of the leading coefficient.
c) If the highest power of x in f(x) is negative, then as x approaches infinity, the function will also approach the horizontal asymptote.

Without knowing the specific function, we cannot determine which of the above scenarios applies.

2. Vertical asymptote at x = 4:
A vertical asymptote occurs when the function approaches infinity or negative infinity as x approaches a certain value. In this case, we have a vertical asymptote at x = 4. This means that as x approaches 4 from the left or the right, the function will tend towards infinity or negative infinity.

To determine the behavior of the function near the vertical asymptote, we should examine the behavior of the function as x approaches 4 from the left (x < 4) and from the right (x > 4). Depending on the function, it may approach infinity or negative infinity as it gets closer to the vertical asymptote.

Again, without knowing the specific function, we cannot determine the exact behavior.

In summary, having a horizontal asymptote at y = 3 means the function gets closer to this line as x approaches infinity. Having a vertical asymptote at x = 4 means the function approaches infinity or negative infinity as x approaches 4. The specific behavior of the function near these asymptotes depends on the form and characteristics of the function itself.

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