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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 2.51a
Chapter 2, Problem 2.51a

For any real number x, the floor function (or greatest integer function) ⌊x⌋ is the greatest integer less than or equal to x (see figure).


a. Compute lim x→−1^− ⌊x⌋, lim x→−1^+ ⌊x⌋,lim x→2^− ⌊x⌋, and lim x→2^+ ⌊x⌋.

Verified step by step guidance
1
Understand the floor function \( \lfloor x \rfloor \), which returns the greatest integer less than or equal to \( x \).
For \( \lim_{x \to -1^-} \lfloor x \rfloor \), consider values of \( x \) approaching \(-1\) from the left. The floor function will return \(-2\) for values like \(-1.1\), \(-1.5\), etc.
For \( \lim_{x \to -1^+} \lfloor x \rfloor \), consider values of \( x \) approaching \(-1\) from the right. The floor function will return \(-1\) for values like \(-0.9\), \(-0.5\), etc.
For \( \lim_{x \to 2^-} \lfloor x \rfloor \), consider values of \( x \) approaching \(2\) from the left. The floor function will return \(1\) for values like \(1.9\), \(1.5\), etc.
For \( \lim_{x \to 2^+} \lfloor x \rfloor \), consider values of \( x \) approaching \(2\) from the right. The floor function will return \(2\) for values like \(2.1\), \(2.5\), etc.

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Key Concepts

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

Floor Function

The floor function, denoted as ⌊x⌋, returns the largest integer that is less than or equal to a given real number x. For example, ⌊3.7⌋ equals 3, while ⌊-2.3⌋ equals -3. Understanding this function is crucial for evaluating limits involving discontinuities at integer values.
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Properties of Functions

Limits from the Left and Right

In calculus, the limit of a function as x approaches a certain value can be evaluated from the left (denoted as lim x→c^−) or from the right (denoted as lim x→c^+). These one-sided limits help determine the behavior of functions at points of discontinuity, which is essential for analyzing the floor function at integer boundaries.
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One-Sided Limits

Discontinuity

A function is said to be discontinuous at a point if there is a sudden jump in its value. The floor function exhibits jump discontinuities at integer values, where the output changes abruptly. Recognizing these discontinuities is vital for correctly computing limits at points where the floor function is evaluated.
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Determine Continuity Algebraically
Related Practice
Textbook Question

Complete the following sentences in terms of a limit.


a. A function is continuous from the left at a if _____.

Textbook Question

Suppose the rental cost for a snowboard is \$25 for the first day (or any part of the first day) plus \$15 for each additional day (or any part of a day).

e. For what values of t is f continuous? Explain.

Textbook Question

Evaluate limx→∞f(x){\(\displaystyle\[\lim\)_{x\(\to\]\infty\)}{f(x)}} andlimx→−∞f(x){\(\displaystyle\]\lim\)_{x\(\to\)-\(\infty\)}{f(x)}}.


f(x)=1−e−2xf\(\left\)(x\(\right\))=1-e^{-2x}

Textbook Question

Determine the following limits.


a. limx→1+x−3x2−5x+4{\(\displaystyle\]\lim\)_{x\(\to\)1^{+}}\(\frac{x-3}{\sqrt{x^2-5x+4}\)}}

Textbook Question

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

lim x→1^− f(x)

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Textbook Question

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x) = (x2 − 9)/(x(x−3))