1/21/2024 0 Comments Discontinuity calculus examplesHowever, if we factor the numerator and denominator, we can cancel the factors (x-2) from both numerator and denominator. Solution: The function f(x) = (x^2 - 4) / (x - 2) is not defined for x = 2. This is a point discontinuity.Įxample 4: Determine the type of discontinuity of the function f(x) = (x^2 - 4) / (x - 2) at x = 2. At this point, the function has a hole in it. Solution: The function f(x) = sqrt(x^2 - 4) is not defined for x = 2. This is an essential discontinuity.Įxample 3: Determine the type of discontinuity of the function f(x) = sqrt(x^2 - 4) at x = 2. At this point, the function has a hole in it and the limit does not exist. As x approaches 0, the function oscillates between positive and negative values, so the left and right limits do not exist. Solution: The function f(x) = x^2 sin(1/x) is not defined for x=0. This is an essential discontinuity, as the function approaches closer and closer to a certain value but never reaches it.Įxample 2: Determine the type of discontinuity of the function f(x) = x^2 sin(1/x) at x = 0 Solution: The function f(x) = 1/x has a vertical asymptote at x = 0. Here are some examples to help you practice:Įxample 1: Determine the type of discontinuity of the function f(x) = 1/x at x = 0. Continuity is a fundamental concept in Calculus, and by understanding how to identify and analyze different types of discontinuities, you will be well-prepared to succeed in your studies of Calculus. You can also try creating your own examples and working through them to get a better understanding of the process. To practice identifying these types of discontinuities, you can try working through examples from your textbook, online resources such as Khan Academy, or by using interactive tools such as graphing calculators or computer programs. Also, it's worth noting that, in real life, a discontinuity may have a physical meaning for instance, a jump discontinuity in a temperature curve may indicate a change of phase. It's important to note that these types of discontinuities are not mutually exclusive and some functions may have more than one type of discontinuity. This can be visualized as a hole in the graph of the function at the discontinuity point that can be filled in. These common factors can be canceled, making the discontinuity "removable". Removable discontinuities occur when a function is a rational expression with common factors in the numerator and denominator. The graph of a function with an essential discontinuity would have a vertical line at the discontinuity point. This can be visualized as the function approaching closer and closer to a certain value, but never reaching it. This can be visualized as a small gap in the graph of the function at the discontinuity point.Įssential discontinuities occur when the curve of a function has a vertical asymptote, also known as an infinite discontinuity. Point discontinuities occur when the function has a "hole" in it at a certain point, meaning that the function has a value that is "off the curve". The graph of a function with a jump discontinuity would look like a step function, with the function abruptly jumping from one value to another at the discontinuity point. This can be visualized as a break in the physical continuity of the graph at a certain point. Jump discontinuities occur when the left and right-handed limits of a function are not equal, resulting in the double-handed limit not existing (DNE). Now for a closer look at discontinuities.
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