Exploring the Concept of Vertical Asymptotes: Understanding its Definition and Behavior

Vertical Asymptote at x = 5

To understand the concept of a vertical asymptote at x = 5, let’s start with the definition of an asymptote

To understand the concept of a vertical asymptote at x = 5, let’s start with the definition of an asymptote. In mathematics, an asymptote is a line that a curve approaches but never touches or crosses.

In this case, we are specifically looking at a vertical asymptote. A vertical asymptote occurs when the values of a function approach positive or negative infinity as x approaches a specific value.

For a vertical asymptote at x = 5, it means that as x gets closer and closer to 5, the values of the function approach positive or negative infinity. This indicates that there is a vertical line at x = 5 that the curve gets arbitrarily close to but never intersects.

To determine why this occurs, we need to look at the behavior of the function as x approaches 5 from the left and right.

For example, let’s consider the function f(x) = 1/(x-5). As x approaches 5 from the left side (x < 5), the denominator (x - 5) approaches 0, but never actually reaches it. However, the reciprocal of a very small positive or negative number becomes increasingly larger, tending towards positive or negative infinity. Therefore, the function approaches infinity as x approaches 5 from the left side. Similarly, as x approaches 5 from the right side (x > 5), the denominator (x – 5) becomes increasingly small, but never reaches 0. Again, the reciprocal of a very small positive or negative number becomes increasingly larger, approaching positive or negative infinity. Thus, the function also approaches infinity as x approaches 5 from the right side.

Therefore, we can conclude that there is a vertical asymptote at x = 5 for the function f(x) = 1/(x-5).

It is worth mentioning that a vertical asymptote can also occur for other types of functions such as rational functions, logarithmic functions, and trigonometric functions, but the approach to finding the vertical asymptote might be a bit different for each function.

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