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- A removable discontinuity appears as a hole in the graph of a function. It occurs when both left-hand and right-hand limits exist and are equal, but the function value at that point is either not defined or does not match the limit. To remove the discontinuity, redefine the function so that it equals the limit at that point.
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What is a removable discontinuity?
How do you determine a removable discontinuity?
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Does a graph of a function have a 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\).
What is a Removable Discontinuity Example? A function y = f(x) has a removable discontinuity at x = a when limₓ → ₐ f(x) ≠ f(a). For example, f(x) = (x 2 - 9) / (x - 3). Then limₓ → ₃ f(x) = limₓ → ₃ [(x -3)(x+3)] / (x - 3) = limₓ → ₃ (x + 3) = 3 + 3 = 6.
Recall that a function f (x) is continuous at a if. lim x→a f (x) = f (a), which can be divided into three conditions: C1: lim x→a f (x) exists. C2: f (a) is defined. C3: C1 = C2. A removable discontinuity occurs when C1 is satisfied, but at least one of C2 or C3 is violated.
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.
Intuitively, it has a removable discontinuity because if you just filled in the hole in the graph, the function would be continuous at \(p\). In other words, removing the discontinuity means changing just one point on the graph.
Since in many applications h is a continuous function, this theorem is often called the limit law for removable discontinuities. This limit law is very useful for efficiently evaluating limits of rational functions and functions like them. For example, consider the function f(x) = x2 x.
In simpler terms, a removable discontinuity happens when there is a hole in the graph of a function, but if you were to fill in that hole with a single point, the function would become continuous at that point. An example of a removable discontinuity can be seen in the function f (x) = (x^2 – 4)/ (x – 2).
