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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 2.9b
Chapter 2, Problem 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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1
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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
Related Practice
Textbook Question

Complete the following steps for the given functions. 


a. Find the slant asymptote of ff.


f(x)=4x3+4x2+7x+4x2+1f\(\left\)(x\(\right\))=\(\frac{4x^3+4x^2+7x+4}{x^2+1}\)

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Textbook Question

Complete the following steps for the given functions. 


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


f(x)=4x3+4x2+7x+4x2+1f\(\left\)(x\(\right\))=\(\frac{4x^3+4x^2+7x+4}{x^2+1}\)

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Textbook Question

Tangent lines with zero slope


a. Graph the function f(x)=x^2−4x+3.

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Textbook Question

Complete the following steps for the given functions. 


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


f(x)=3x2−2x+53x+4f\(\left\)(x\(\right\))=\(\frac{3x^2-2x+5}{3x+4}\)

346
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Textbook Question

Graph the function f(x)=e^−x / x(x+2)^2 using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits.


b. lim x→−2 f(x)

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Textbook Question

Graph the function f(x)=e^−x / x(x+2)^2 using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits.


a. lim x→−2^+ f(x)

354
views