When do vertical asymptotes occur in a function's graph?

Vertical asymptotes occur when a function approaches infinity or negative infinity as the input (x-value) approaches a certain value. This behavior is typically observed in rational functions where the denominator of the function becomes zero, while the numerator does not simultaneously equal zero at those points.

Thus, when a factor in the denominator equals zero, it indicates that the function cannot be defined at that x-value, leading to a vertical asymptote on the graph. At this point, the function's value tends to infinitely large or small values, resulting in a vertical line in the graph indicating that the function cannot take any value at this input.

In contrast, if a factor in the numerator equals zero or both the numerator and denominator equal zero, this situation would not typically create a vertical asymptote. Instead, other behaviors, such as holes in the graph or a need for further analysis (like removable discontinuities), might occur. Understanding these conditions is essential when analyzing rational functions for vertical asymptotes.