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What is an example of a removable discontinuity?
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The removable discontinuity of a graph is a point where it has a hole. A function f(x) is has a removable discontinuity at x = a if its limit exists at x = a but it is not equal to f(a). Learn more about removable discontinuity along with examples.
What is Removable Discontinuity? A removable discontinuity occurs at a point on a function where the function is not defined, yet the limit as we approach that point exists. Mathematically, for a function \(f(x)\) with a removable discontinuity at \(x = a\), we can say: - \(f(a)\) is undefined or does not equal the limit as \(x\) approaches \(a\).
Nov 21, 2023 · A removable discontinuity is an x-value in a function for which the two one-sided limits are equal and finite but either the function is undefined at c or the limit at c is not equal to f(c).
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- If there is a common factor in the numerator and the denominator of a rational function, set that factor equal to zero and solve for x. Plug the x-...
- If the break in the function can be plugged with a single point, it is removable. If not, it is non-removable. Holes and point discontinuities are...
- A removable discontinuity is a break in a function that can be plugged with a single point. The two one-sided limits at x = c are equal to each oth...
- Quick Overview
- Jump Discontinuities
- Infinite Discontinuities
- Removable Discontinuities
- Removable Discontinuities Can Be Fixed
- Endpoint Discontinuities
- Mixed Discontinuities
Discontinuities can be classified as jump, infinite, removable, endpoint, or mixed.Removable discontinuities are characterized by the fact that the limitexists.Removable discontinuities can be "fixed" by re-defining the function.The other types of discontinuities are characterized by the fact that the limit does not exist. Specifically,The graph of below shows a function that is discontinuous at . In this graph, you can easily see that The function is approaching different values depending on the direction is coming from. When this happens, we say the function has a jump discontinuity at .
The graph below shows a function that is discontinuous at . The arrows on the function indicate it will grow infinitely large as approaches . Since the function doesn't approach a particular finite value, the limit does not exist. This is an infinite discontinuity. The following two graphs are also examples of infinite discontinuities at . Notice t...
In the graphs below, there is a hole in the function at . These holes are called removable discontinuities Notice that for both graphs, even though there are holes at , the limit value at exists.
Removable discontinuities can be fixed by redefining the function, as shown in the following example.
When a function is defined on an interval with a closed endpoint, the limit cannot exist at that endpoint. This is because the limit has to examine the function values as approaches from both sides. For example, consider finding (see the graph below). Note that is the left-endpoint of the functions domain: , and the function is technically not cont...
Consider the graph shown below. The function is obviously discontinuous at . From the left, the function has an infinite discontinuity, but from the right, the discontinuity is removable. Since there is more than one reason why the discontinuity exists, we say this is a mixeddiscontinuity
3 days ago · A real-valued univariate function f=f(x) is said to have a removable discontinuity at a point x_0 in its domain provided that both f(x_0) and lim_(x->x_0)f(x)=L<infty (1) exist while f(x_0)!=L.
Figure 1: A removable discontinuity: the function is continuous everywhere except one point For example, g (x) = sin(x) and h 1−cos x x are defined for x = 0, but x both functions have removable discontinuities. This is not obvious at all, but we will learn later that: sin x 1 − cos x lim = 1 and lim = 0. x→0 x x→0 x
A removable discontinuity is a discontinuity that results when the limit of a function exists but is not equal to the value of the function at the given point. It is referred to as removable because the function can be re-defined as a piecewise function such that it becomes continuous.
