Problem 90
Question
Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If \(\lim _{x \rightarrow 0} f(x)=4\) and \(\lim _{x \rightarrow 0} g(x)=0\), then \(\lim _{x \rightarrow 0} f(x) g(x)=0\).
Step-by-Step Solution
Verified Answer
The statement is true. Based on the product rule for limits, we have \(\lim_{x \rightarrow 0} f(x)g(x) = \lim_{x \rightarrow 0} f(x) \cdot \lim_{x \rightarrow 0} g(x)\), which, by substituting the given limit values, results in \(\lim_{x \rightarrow 0} f(x)g(x) = 4 \cdot 0 = 0\).
1Step 1: Analyze the Given Information
We are given the respective limits of two functions:
1. \(\lim_{x \rightarrow 0} f(x) = 4\)
2. \(\lim_{x \rightarrow 0} g(x) = 0\)
We are asked to evaluate whether:
\(\lim_{x \rightarrow 0} f(x)g(x) = 0\)
2Step 2: Apply Limit Properties
To determine the limit of the product f(x)g(x) as x approaches 0, we can apply the properties of limits, notably the product rule for limits, which states that:
\(\lim_{x \rightarrow a} f(x)g(x) = \lim_{x \rightarrow a} f(x) \cdot \lim_{x \rightarrow a} g(x)\), provided that both limits exist.
In our case, a = 0, and it is already given that the limits exist.
3Step 3: Calculate the Limit of the Product
Using the product rule, we can compute the limit of the product f(x)g(x) as x approaches 0:
\(\lim_{x \rightarrow 0} f(x)g(x) = \lim_{x \rightarrow 0} f(x) \cdot \lim_{x \rightarrow 0} g(x)\)
We are given the values of these two limits, so we can substitute them:
\(\lim_{x \rightarrow 0} f(x)g(x) = 4 \cdot 0 = 0\)
4Step 4: Compare the Results
Now we can see that the limit of the product f(x)g(x) as x approaches 0 is indeed 0:
\(\lim_{x \rightarrow 0} f(x)g(x) = 0\)
Since the calculated limit matches the value in the given statement, we can conclude that:
The statement is true, because \(\lim_{x \rightarrow 0} f(x)g(x) = 0\), as shown by the product rule for limits.
Key Concepts
Limit PropertiesProduct Rule for LimitsLimit Evaluation
Limit Properties
When studying calculus, understanding limit properties is crucial for analyzing how functions behave as they approach a specific point. Limit properties are rules that allow us to break down complex limit problems into simpler parts. This is particularly useful when we can't directly substitute a value into a function due to indeterminate forms or discontinuities.
Some of the fundamental properties of limits include the sum rule, which allows us to add limits together; the product rule, which deals with multiplying limits; and the quotient rule, which is used when dividing limits. There's also the power rule for when a limit is raised to a power. These properties become tools that help us solve more complex limit problems efficiently and accurately.
Applying the right property correctly is crucial. An incorrect application can lead to wrong answers, as limits do not obey all the algebraic properties that regular numbers do. In the problem discussed, we apply the product rule for limits, which only holds true if the limits of the individual functions involved exist independently. As illustrated, this approach simplifies the problem and successfully leads to an accurate solution.
Some of the fundamental properties of limits include the sum rule, which allows us to add limits together; the product rule, which deals with multiplying limits; and the quotient rule, which is used when dividing limits. There's also the power rule for when a limit is raised to a power. These properties become tools that help us solve more complex limit problems efficiently and accurately.
Applying the right property correctly is crucial. An incorrect application can lead to wrong answers, as limits do not obey all the algebraic properties that regular numbers do. In the problem discussed, we apply the product rule for limits, which only holds true if the limits of the individual functions involved exist independently. As illustrated, this approach simplifies the problem and successfully leads to an accurate solution.
Product Rule for Limits
A key concept in calculus is the product rule for limits, which is used when trying to find the limit of a product of two functions. This rule states that the limit of a product is equal to the product of the limits, formally expressed as \( \lim_{x \rightarrow a} f(x)g(x) = \lim_{x \rightarrow a} f(x) \cdot \lim_{x \rightarrow a} g(x)\), given that the individual limits exist and are finite.
This principle allows us to break down a potentially complicated limit of a product into more manageable parts. When using the product rule, it is vital to first confirm that the individual limits of \(f(x)\) and \(g(x)\) as \(x\) approaches \(a\) exist. In the provided exercise, since we know the individual limits of \(f(x)\) and \(g(x)\) as \(x\) approaches 0, we can apply the product rule to find that indeed, the limit of the product is 0. The product rule elegantly handles situations where direct substitution is not possible and is a testament to the power of limit properties in calculus.
This principle allows us to break down a potentially complicated limit of a product into more manageable parts. When using the product rule, it is vital to first confirm that the individual limits of \(f(x)\) and \(g(x)\) as \(x\) approaches \(a\) exist. In the provided exercise, since we know the individual limits of \(f(x)\) and \(g(x)\) as \(x\) approaches 0, we can apply the product rule to find that indeed, the limit of the product is 0. The product rule elegantly handles situations where direct substitution is not possible and is a testament to the power of limit properties in calculus.
Limit Evaluation
Limit evaluation is the process of determining the value that a function approaches as the input approaches a certain value. It is a core aspect of calculus that deals with the behavior of functions at specific points, which can be finite or at infinity. The accurate evaluation of limits often relies on applying the aforementioned limit properties effectively.
When evaluating limits, it's important to recognize when to perform algebraic simplifications or when to apply specific limit rules. For instance, understanding when you can apply L'Hôpital's rule, which is used to evaluate limits that result in indeterminate forms like 0/0 or ∞/∞, is essential. Moreover, in the case of piecewise functions or functions with discontinuities, the evaluation must consider different approaches towards the point of interest.
In the example at hand, we evaluate the limit of a product by multiplying the individual limits of the functions \(f(x)\) and \(g(x)\), which is a straightforward application of the product rule. This leads to a clear and correct conclusion, showcasing that thorough understanding and correct use of limit evaluation techniques is critical to success in solving calculus problems.
When evaluating limits, it's important to recognize when to perform algebraic simplifications or when to apply specific limit rules. For instance, understanding when you can apply L'Hôpital's rule, which is used to evaluate limits that result in indeterminate forms like 0/0 or ∞/∞, is essential. Moreover, in the case of piecewise functions or functions with discontinuities, the evaluation must consider different approaches towards the point of interest.
In the example at hand, we evaluate the limit of a product by multiplying the individual limits of the functions \(f(x)\) and \(g(x)\), which is a straightforward application of the product rule. This leads to a clear and correct conclusion, showcasing that thorough understanding and correct use of limit evaluation techniques is critical to success in solving calculus problems.
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