A hole is a point on the graph where the value of the function is not defined. If the numerator and denominator of a rational function have a common factor, they will cancel when simplifying. The cancelled value creates a hole in the graph.
A hole on a graph looks like a hollow circle. It represents the fact that the function approaches the point, but is not actually defined on that precise value. The reason why this function is not defined at is because is not in the domain of the function.
Holes occur when factors from the numerator and the denominator cancel. When a factor in the denominator does not cancel, it produces a vertical asymptote.
Holes are caused by the presence of an identical zero or root in both the numerator and denominator of a rational function. Rational functions have zeros (roots), points where the graph crosses the x-axis, or f(x) = 0, just like polynomial functions.
Vertical asymptotes are "holes" in the graph where the function cannot have a value. They stand for places where the x-value is not allowed. Specifically, the denominator of a rational function cannot be equal to zero. Any value of x that would make the denominator equal to zero is a vertical asymptote.
An asymptote is a line that a graph approaches without touching. Similarly, horizontal asymptotes occur because y can come close to a value, but can never equal that value. In the previous graph, there is no value of x for which y = 0 ( ≠ 0), but as x gets very large or very small, y comes close to 0.
Infinite (essential) discontinuity
Vertical asymptotes are only points of discontinuity when the graph exists on both sides of the asymptote. The graph below shows a vertical asymptote that makes the graph discontinuous, because the function exists on both sides of the vertical asymptote.
Removable Discontinuities
If there is a removable discontinuity (also known as a 'hole') in the curve of the graph at x = c, then the limit does exist on the graph of a function.
If there is no common factor at both numerator and denominator, there is no hole for the rational function. Case 2 : If there is a common factor at both numerator and denominator, there is a hole for the rational function.
A rational function doesn't have to have a vertical or horizontal asymptote.
If the numerator and denominator of a rational expression have a shared factor that cancels, then the graph of the rational function will have a hole in it. This is distinct from the graph having a gap due to a vertical asymptote.
The graph of a rational function usually has vertical asymptotes where the denominator equals 0. However, the graph of a rational function will have a hole when a value of x causes both the numerator and the denominator to equal 0. This occurs when there is a common factor in the numerator and denominator.
A continuous function can be represented by a graph without holes or breaks. A function whose graph has holes is a discontinuous function.
A rational function cannot cross a vertical asymptote because it would be dividing by zero. Horizontal asymptotes occur when the x-values get very large in the positive or negative direction.
Some functions have asymptotes because the denominator equals zero for a particular value of x or because the denominator increases faster than the numerator as x increases.
Since a linear function is continuous everywhere, linear functions do not have any vertical asymptotes.
Given a rational function if a number causes the denominator and the numerator to be 0 then both the numerator and denominator can be factored and the common zero can be cancelled out. This means there is a hole in the function at this point.
Explanation: A vertical asymptote usually corresponds to a 'hole' in the domain, and a horizontal asymptote often corresponds to a 'hole' in the range, but those are the only correspondences I can think of. Then t(x) has vertical asymptotes at (2k+1)π2 for all k∈Z , but has no 'holes'.
The difference between a "removable discontinuity" and a "vertical asymptote" is that we have a R. discontinuity if the term that makes the denominator of a rational function equal zero for x = a cancels out under the assumption that x is not equal to a. Othewise, if we can't "cancel" it out, it's a vertical asymptote.
A hole is a point on the graph where the value of the function is not defined. If the numerator and denominator of a rational function have a common factor, they will cancel when simplifying. The cancelled value creates a hole in the graph.
Note that a graph can have both a vertical and a slant asymptote, or both a vertical and horizontal asymptote, but it CANNOT have both a horizontal and slant asymptote.