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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 2.9a
Chapter 2, Problem 2.9a

Complete the following sentences in terms of a limit.


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

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Step 1: Understand the concept of continuity from the left. A function is continuous from the left at a point \( a \) if the left-hand limit of the function as \( x \) approaches \( a \) is equal to the function's value at \( a \).
Step 2: Express the left-hand limit mathematically. The left-hand limit of a function \( f(x) \) as \( x \) approaches \( a \) from the left is denoted as \( \lim_{{x \to a^-}} f(x) \).
Step 3: State the condition for left continuity. For the function \( f(x) \) to be continuous from the left at \( a \), the condition \( \lim_{{x \to a^-}} f(x) = f(a) \) must be satisfied.
Step 4: Consider the implications. This means that as \( x \) gets arbitrarily close to \( a \) from values less than \( a \), the function values \( f(x) \) should approach \( f(a) \).
Step 5: Summarize the sentence. A function is continuous from the left at \( a \) if \( \lim_{{x \to a^-}} f(x) = f(a) \).

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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 defined at those points.
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Limits of Rational Functions: Denominator = 0

Continuity

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. This means there are no breaks, jumps, or holes in the graph of the function at that point, ensuring a smooth transition.
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Intro to Continuity

One-Sided Limits

One-sided limits refer to the behavior of a function as it approaches a specific point from one side only, either the left or the right. For a function to be continuous from the left at a point 'a', the left-hand limit must equal the function's value at 'a', indicating that the function approaches a specific value as the input approaches 'a' from the left.
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Related Practice
Textbook Question

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⌋.

Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.


a. The graph of a function can never cross one of its horizontal asymptotes.

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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

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}.


a. Determine its end behavior by analyzing limx→∞cosh(x){\(\displaystyle\[\lim\)_{x\(\to\]\infty\)}{\(\cosh\)(x)}} and limx→−∞cosh(x){\(\displaystyle\]\lim\)_{x\(\to\)-\(\infty\)}{\(\cosh\)(x)}}.

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))

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)