Understanding Vertical Asymptotes | Exploring the Behavior of Functions as x Approaches Certain Values

vertical asymptote

A vertical asymptote is a line that a graph approaches but never touches as the input values (x-values) increase or decrease without bound

A vertical asymptote is a line that a graph approaches but never touches as the input values (x-values) increase or decrease without bound. In other words, it is a vertical line that represents a value that the function cannot attain.

To understand vertical asymptotes, we need to consider the behavior of a function as it approaches certain values of x. Let’s use the example of a rational function, which is a fraction of two polynomials. Consider the function f(x) = (x^2 – 1) / (x – 1).

To determine if there is a vertical asymptote, we need to see if there is a value of x that makes the denominator of the fraction equal to zero. In this case, x = 1 is the value that makes the denominator zero.

Now, for a vertical asymptote to exist at x = a, the function must have one of these three behaviors as x approaches a:
1. The function approaches positive infinity: The function values increase without bound as x gets closer to a.
2. The function approaches negative infinity: The function values decrease without bound as x gets closer to a.
3. The function approaches either positive or negative infinity depending on whether we approach a from the left or the right.

In our example, as x approaches 1, the numerator (x^2 – 1) approaches 0, while the denominator (x – 1) approaches 0 as well. However, since the numerator approaches 0 faster than the denominator does, the function approaches positive infinity as x gets closer to 1 from the left side, and it approaches negative infinity as x gets closer to 1 from the right side. Therefore, we have a vertical asymptote at x = 1.

This means that the graph of the function f(x) = (x^2 – 1) / (x – 1) approaches the line x = 1 but never touches it. It becomes steeper and steeper as x approaches 1, but it does not cross the line.

Vertical asymptotes can also exist for other types of functions, such as exponential or logarithmic functions, but the concept remains the same. It represents a value that the function cannot attain as the x-values approach it.

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