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Description
When you have a piecewise-defined function that includes a variable, you can find the value of that variable that makes the function continuous. Just plug in the value where the "split" occurs, then set the remaining two functions equal to one another and solve for the remaining variable. You're basically just finding the value that makes the left- and right-hand limits equal, which would make the function continuous.
Description
If the graph of a function as a hole at a single point, it's called a removable discontinuity because the discontinuity can be "removed" just by redefining the value of the function at that singular point. Any rational function in which you can cancel the same factor from the numerator and denominator, has a removable discontinuity.
Description
The late 20th century produced a sinister euphemism: "ethnic cleansing." This program concludes a comprehensive survey of genocide by looking at the most recent examples in Iraq, Iran, and Turkey; Burundi and Rwanda; the former Yugoslavia; Indonesia and East Timor; and Chechnya. The role and efforts of the United Nations are discussed as well as what the future holds in trying to prevent genocide. Among many scholars, experts, and survivors interviewed...
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"Continuity & Change in the American Family engages students with issues they see every day in the news, while providing them with a comprehensive description of the social demography of the American family. Understanding ever-changing family systems and patterns requires taking the pulse of contemporary family life from time to time. This book paints a portrait of family continuity and change in the latter half of the 20th century, with focus on...
Description
Sometimes you'll be asked to find various limits of a function defined by a "crazy graph". The trick is to understand that the limit is just the value the function approaches as you trace your finger along the graph toward the limit value. You may find that the left- and right-hand limits of a function are different at some points, and that the value of the function at a point is not always equal to the limit of the function there.
Description
To use factoring to solve for the limit of a rational function, just factor the numerator and denominator completely, then see if you can cancel common factors from the fraction. This might simplify the fraction to the point where you can use substitution with the remaining function to find the limit.
Description
The squeeze theorem is another method we can use to find the limit of a function. If we can show that the limit of a function at a point is always greater than or equal to some value and less than or equal to that same value, than by the squeeze theorem we can prove that the limit of the function a that point has the same value.
16) Domain of a Multivariable Function: Example 2 :Calculus-Partial Derivatives: Limits and Continuity
Description
The domain of a multivariable function will be defined in terms of three-dimensional space (3D, or R3), whereas the domain of a single variable function could be defined in terms of two-dimensional space (2D, or R).
Description
The general limit of a function only exists when the left-hand limit exists, the right-hand limit exists, and the left- and right-hand limits are equal to one another. Even when the general limit doesn't exist because one of these conditions isn't met, the one-sided limits (the left-hand limit and/or the right-hand limit), can still exist independently.
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