Understanding Vertical Asymptotes in Mathematics | Explained and Demonstrated

Vertical asymptote at x = a

In the context of mathematics, a vertical asymptote is a line that a function approaches but never intersects as the input values (x-values) approach a certain value

In the context of mathematics, a vertical asymptote is a line that a function approaches but never intersects as the input values (x-values) approach a certain value. Specifically, if a function f(x) has a vertical asymptote at x = a, it means that the graph of the function approaches the line x = a as x tends to a specific value.

To understand this concept, let’s consider an example. Suppose we have the function f(x) = 1 / (x – 3). This function has a vertical asymptote at x = 3.

To determine the vertical asymptote, we need to see what happens to the function as x approaches the value where the asymptote is located. In our case, as x gets closer and closer to 3, the denominator of the function (x – 3) gets smaller and smaller. When the denominator becomes zero, the whole fraction becomes undefined. So, we can say that x = 3 is a value that makes the function undefined.

However, it’s important to note that just because a value makes the function undefined doesn’t necessarily mean it is a vertical asymptote. For it to be a vertical asymptote, the function must approach the line x = 3 without intersecting it.

In this example, as x approaches 3 from the left side (x < 3), the function's values become increasingly negative. Similarly, as x approaches 3 from the right side (x > 3), the function’s values become increasingly positive. This behavior implies that the graph of the function approaches the vertical line x = 3 but never intersects it.

Therefore, we can conclude that the function f(x) = 1 / (x – 3) has a vertical asymptote at x = 3.

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