A removable discontinuity is a point where a function is not defined, but the function can be made continuous by assigning a value at that point. It is a type of discontinuity that can be 'removed' by redefining the function to be continuous at that point.
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A removable discontinuity occurs when the function is not defined at a particular point, but the limit of the function as it approaches that point exists.
Removable discontinuities are often found in rational functions when the numerator and denominator have a common factor that can be canceled out.
The limit of a function with a removable discontinuity can be found by applying L'Hôpital's rule or by redefining the function to be continuous at the point of discontinuity.
Continuity is an important property of functions, as it ensures that the function behaves in a predictable way and that it can be integrated or differentiated.
Identifying and handling removable discontinuities is a crucial skill in calculus, as it allows for the proper analysis of function behavior and the computation of limits.
Review Questions
Explain how a removable discontinuity differs from other types of discontinuities, and describe the conditions under which a discontinuity is considered removable.
A removable discontinuity is distinct from other types of discontinuities, such as jump discontinuities or asymptotic discontinuities, in that the function can be made continuous by simply assigning a value at the point of discontinuity. For a discontinuity to be considered removable, the limit of the function as it approaches the point of discontinuity must exist, even though the function is not defined at that point. This means that the function can be redefined to be continuous at the point of discontinuity without altering the function's behavior elsewhere.
Describe the relationship between removable discontinuities and rational functions, and explain how identifying and handling these discontinuities can be useful in the analysis of rational functions.
Removable discontinuities are often found in rational functions when the numerator and denominator have a common factor that can be canceled out. By identifying and handling these removable discontinuities, you can gain important insights into the behavior of rational functions. For example, you can determine the points at which the function is not defined, find the limits of the function as it approaches these points, and ultimately, understand the overall continuity and behavior of the rational function.
Explain how the concept of continuity is related to removable discontinuities, and discuss the importance of continuity in the context of calculus and the analysis of function behavior.
Continuity is a fundamental property of functions, and it is closely tied to the concept of removable discontinuities. A function with a removable discontinuity can be made continuous by redefining the function at the point of discontinuity, which ensures that the function behaves in a predictable way. Continuity is essential in calculus, as it allows for the proper computation of limits, derivatives, and integrals. By identifying and handling removable discontinuities, you can ensure the continuity of a function, which is crucial for the accurate analysis of function behavior and the application of calculus techniques.
A function is continuous at a point if the function's value at that point is equal to the limit of the function as the input approaches that point from both sides.