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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
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Capitolo 2, Problema 2.9b

Complete the following sentences in terms of a limit.


b. A function is continuous from the right at a if _____ .

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To understand continuity from the right at a point 'a', we need to consider the behavior of the function as it approaches 'a' from the right side.
The formal definition involves limits. Specifically, a function f(x) is continuous from the right at a point 'a' if the right-hand limit of f(x) as x approaches 'a' is equal to the function value at 'a'.
In mathematical terms, this can be expressed as: limx→a+f(x)=f(a)
This means that as x gets arbitrarily close to 'a' from values greater than 'a', the function values f(x) should approach f(a).
If this condition is satisfied, we say that the function is continuous from the right at the point 'a'.

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Limit of a Function

The limit of a function describes the value that the function approaches as the input approaches a certain point. It is a fundamental concept in calculus that helps in understanding the behavior of functions near specific points, especially when they are not explicitly defined at those points.
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Limits of Rational Functions: Denominator = 0

Continuity

A function is continuous at a point if there is no interruption in its graph at that point. This means that the limit of the function as it approaches the point from both sides equals the function's value at that point. Continuity is essential for ensuring that small changes in input result in small changes in output.
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Intro to Continuity

Right-Hand Limit

The right-hand limit of a function at a point 'a' refers to the value that the function approaches as the input approaches 'a' from the right (values greater than 'a'). For a function to be continuous from the right at 'a', the right-hand limit must equal the function's value at 'a'.
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One-Sided Limits
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Domanda del libro di testo

The hyperbolic cosine function, denoted cosh(x)\(\cosh\)\(\left\)(x\(\right\)), is used to model the shape of a hanging cable (a telephone wire, for example). It is defined as cosh(x)=ex+e−x2\(\cosh\)\(\left\)(x\(\right\))=\(\frac{e^{x}\)+e^{-x}}{2}.


b. Evaluate cosh(0)\(\cosh\)\(\left\)(0\(\right\)). Use symmetry and part (a) to sketch a plausible graph for y=cosh(x)y=\(\cosh\)\(\left\)(x\(\right\)).

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Complete the following steps for the given functions. 


b. Find the vertical asymptotes of ff (if any).


f(x)=x2−3x+6f\(\left\)(x\(\right\))=\(\frac{x^2-3}{x+6}\)

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The graph of f in the figure has vertical asymptotes at x=1 and x=2. Analyze the following limits. <IMAGE>

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Use analytic methods to find the value of lim x→π/4 cos 2x / cos x − sin x.

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Determine the following limits.


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