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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 71a
Chapter 2, Problem 71a

Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers.
If limx→a f(x) = L, then f(a)=L.

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Step 1: Understand the statement. The statement claims that if the limit of a function \( f(x) \) as \( x \) approaches \( a \) is \( L \), then the function value at \( a \), \( f(a) \), must also be \( L \).
Step 2: Recall the definition of a limit. The limit \( \lim_{x \to a} f(x) = L \) means that as \( x \) gets arbitrarily close to \( a \), \( f(x) \) gets arbitrarily close to \( L \).
Step 3: Consider the concept of continuity. A function \( f(x) \) is continuous at \( x = a \) if \( \lim_{x \to a} f(x) = f(a) \).
Step 4: Identify a counterexample. Consider the function \( f(x) = \frac{\sin(x)}{x} \) for \( x \neq 0 \) and \( f(0) = 1 \). The limit \( \lim_{x \to 0} \frac{\sin(x)}{x} = 1 \), but \( f(0) = 1 \).
Step 5: Conclude the analysis. The statement is not necessarily true because a function can have a limit at a point without being continuous at that point, meaning \( f(a) \) does not have to equal \( L \).

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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 behavior of the function as the input approaches a certain value. Specifically, limx→a f(x) = L means that as x gets arbitrarily close to a, the values of f(x) approach L. This concept is fundamental in calculus as it helps in understanding continuity and the behavior of functions near specific points.
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Continuity

A function is continuous at a point a if the limit of the function as x approaches a equals the function's value at that point, i.e., limx→a f(x) = f(a). This means there are no breaks, jumps, or holes in the graph of the function at that point. Understanding continuity is crucial for evaluating the truth of the statement regarding limits and function values.
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Counterexample

A counterexample is a specific case that disproves a general statement. In the context of limits, if we find a function where limx→a f(x) = L but f(a) ≠ L, it serves as a counterexample to the claim that the limit implies the function's value at that point. This concept is essential for critical thinking and validating mathematical statements.
Related Practice
Textbook Question

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim t→2+ |2t − 4|t^2 − 4

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Textbook Question

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a- f(x) and lim x→a+ f(x).

f(x) = (x2 − 4x + 3) / (x − 1)

Textbook Question

Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers.

The limit lim x→a f(x) / g(x) does not exist if g(a)=0.

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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 − 4x + 3) / (x − 1)

Textbook Question

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

f(x) = (√(16x4 + 64x2) + x2) / (2x2 − 4) 

Textbook Question

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a- f(x) and lim x→a+ f(x).

f(x) = (2x3 + 10x2 + 12x) / (x3 + 2x2)

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