When factors from the numerator and denominator cancel, a hole is formed. When a factor in the denominator does not cancel, a vertical asymptote is produced. These are two different effects that can occur together.

Vertical asymptotes occur when a factor in a rational expression's denominator does not cancel with a factor in the numerator. When a factor does not cancel, instead of leaving a hole at that x value, a vertical asymptote exists. At any point where there is no horizontal distance between two lines, gravity has no effect and therefore objects on either side of the point reach their maximum altitude simultaneously.

When calculating the altitude of objects near the surface of the earth, it is reasonable to assume that they are stationary relative to the ground. Therefore, their altitudes will not change over time. Objects far from the surface of the earth, such as those orbiting the planet at high speeds or those located in **outer space**, are free to move up or down depending on how they are oriented with respect to the Earth's gravitational field. For **these reasons**, objects near and far have different elevations at **any given moment**.

When objects near the surface of the earth are moving toward or away from each other, their respective elevations will also be changing. If we were to stop the motion of one of the objects at some point in time, then the object would reach its highest elevation immediately before stopping. Similarly, if we stopped the movement of the second object, then it would reach its highest elevation immediately after stopping.

Set each denominator factor to zero and solve for the variable. This component is a vertical asymptote of the equation if it does not exist in the numerator. If it does exist in the numerator, the equation has a hole. A hole means that there are values of the independent variable at which the function is not defined.

At any input value, a hole appears on the graph of a rational function, causing both the numerator and denominator of the function to be equal to zero. They occur when components in rational functions may be algebraically wiped out. For example, if there were a hole at x = 3, then the function would have no undefined points. A hole can also appear as an output value for a finite-state machine.

Discontinuity that is removable Holes are another name for removable discontinuities.

The term "hole" is also used in mathematics to describe an open subset of a topological space. In set theory, a hole is a set with no elements; more generally, a hole is a non-empty set whose only subsets are itself and its own empty set. A function with a hole is one that is defined on some (or, equivalently, every) element of the domain except one: there is at least one value where the function is not defined. The exception is called the hole of the function. There are several ways to define when two holes are equal; we will use two common ones here. If there is a way to fill each hole with a single value, then the holes are said to be identical. Otherwise, the holes are different.

Holes can appear in various places in mathematical analysis. For example, if we have found all the solutions to an equation within some range, there will usually be one solution which does not fall within this range. This solution is known as the "extremal" solution because it lies outside of the range of values that could possibly be assigned to the variable in question.

As a result, the vertical asymptotes of a function are the x values that create the bottom zero but not the top zero. That is, they're the x values below which the function does not approach a limit from above or below.

A vertical asymptote is a value at which a rational function is undefined, indicating that the value is not in the function's domain. The graph of a rational function will always cross over itself, so there will be values of x for which the function is defined for both positive and negative values of y. However, since the function is only defined for positive values of y, it has a vertical asymptote at which it is defined for only one sign of y.

Domains are the ranges of values of the variable for which the function is defined. A domain can include values for which the function is undefined or not defined. For example, the domain of $\frac{1}{x}$ includes **all non-zero numbers**, while that of $e^x$ does not include zero because it leads to problems with exponentiation rules. Domains can also include points at which the function is not differentiable (where derivatives don't exist). For example, the domain of $f(x) = \sin x$ does not include $\pi$ because we can't differentiate $\sin\pi$. Domains can also exclude **certain values** when those values would make the function undefined.