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Which of the following functions $ f $ has a removable discontinuity at $ a $? If the discontinuity is removable, find a function $ g $ that agrees with $ f $ for $ x \neq a $ and is continuous at $ a $.
(a) $ f(x) = \dfrac{x^4 -1}{x - 1}$, $ a = 1 $(b) $ f(x) = \dfrac{x^3 - x^2 - 2x}{x - 2} $, $ a = 2 $(c) $ f(x) = [ \sin x ] $, $ a = \pi $
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Calculus 1 / AB
Chapter 2
Limits and Derivatives
Section 5
Continuity
Limits
Derivatives
Brandon C.
January 15, 2023
For (c), isn't the limit as x --> pi ^ + equal to zero? The sine function is equal to -1 at 3pi/2 unless I'm misunderstanding something.
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University of Michigan - Ann Arbor
University of Nottingham
Lectures
04:40
In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.
In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.
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