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Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain.

$ N(r) = \tan^{-1}(1 + e^{-r^2}) $

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Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 5

Continuity

Limits

Derivatives

Missouri State University

Harvey Mudd College

University of Nottingham

Idaho State University

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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Explain, using Theorems 4,…

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$25-32$ Explain, using The…

this is problem number thirty two of the sewer calculus eighth edition section two point five. Explain using terms four, five, seven and nine where the function is continuous. That every number in its domain state to state that I mean this function and our is equal to inverse tangent or arc tangent of the quantity one plus eed to the negative are squared. So we refer to our therms. We have a function within a function. This has to do with your nine. So as long as the function on the inside is continuous for its demon and the function, yes, it is also continues on its domain. Then the composite function of the two is also continuous. Within the Arc Tangent function we have on exponential function, this function is definitely continuous. That's so mean and in fact that's something his all real numbers since our is allowed to be any number. So there are no domain restrictions and then they are tendon function. This inverse tangent function is a form of it is a example of a triple trigon metric function that is continuous on all reals. That's the domain of this specific. You're gonna mention function here in seven states that any treatment function is continuous on its doing, but specifically our attention. Its domain specifically is negative. Infinity to infinity. So all rials on and as we discussed, both of these have no Domaine restrictions. So since this is continuous and all rials and our attention function also continues on Honoria, all rials, no domain restrictions means that the combined domain for this composite function it's going to be all real numbers from negative infinity. That's your final answer.

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