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Chapter 2: Problem 207

For the following exercises, suppose that $$\lim _{x \rightarrow a} f(x)=L$$ and $$\lim _{x \rightarrow a} g(x)=M$$ both exist. Use the precise definition of limits to prove the following limit laws: $$\lim _{x \rightarrow a}[f(x) g(x)]=L M$$ (Hint: \(|f(x) g(x)-L M|=\) \(|f(x) g(x)-f(x) M+f(x) M-L M| \leq|f(x) \| g(x)-M|+|M| f(x)-L |\) )

Short Answer

Expert verified
Proven by definition and inequalities: \( \lim_{x \to a}[f(x)g(x)] = LM \).

Step by step solution

01

Understand the Goal

We need to prove that \( \lim_{x \to a}[f(x) g(x)] = LM \), using the hint provided and the precise definition of a limit.
02

Apply the Precise Definition of a Limit

According to the definition, for any \( \epsilon > 0 \), there exist \( \delta_1, \delta_2 > 0 \) such that:1. \( |f(x) - L| < \frac{\epsilon}{2(|M|+1)} \) whenever \( 0 < |x-a| < \delta_1 \)2. \( |g(x) - M| < \frac{\epsilon}{2|L|} \) whenever \( 0 < |x-a| < \delta_2 \)
03

Choose \(\delta\) Appropriately

Let \(\delta = \min(\delta_1, \delta_2)\). This ensures that both \( |f(x) - L| < \frac{\epsilon}{2(|M|+1)} \) and \( |g(x) - M| < \frac{\epsilon}{2|L|} \) hold for \( 0 < |x-a| < \delta \).
04

Use the Hint to Manipulate the Expression

From the hint, we can say:\[|f(x)g(x) - LM| = |f(x)g(x) - f(x)M + f(x)M - LM| \leq |f(x)||g(x) - M| + |M||f(x) - L|\]
05

Substitute the Limits and Bounds

Using the fact that \( |f(x)| \leq |L| + \frac{\epsilon}{2(|M|+1)} \) when \( x \) is sufficiently close to \( a \) (due to \(|f(x)-L|<\frac{\epsilon}{2(|M|+1)}\)):\[|f(x)||g(x) - M| < (|L| + \frac{\epsilon}{2(|M|+1)})\frac{\epsilon}{2|L|}\]This term can be made smaller than \( \frac{\epsilon}{2} \) for sufficiently small \( \epsilon \).
06

Show \(|M||f(x) - L|\) is Small

Similarly, showing that \(|M||f(x) - L| < \frac{\epsilon}{2}\) is possible since we selected \(|f(x) - L|<\frac{\epsilon}{2|M|}\).
07

Conclude with the Precise Definition

Thus, combining both results:\[|f(x)g(x) - LM| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon\]Therefore, \( \lim_{x \to a} [f(x)g(x)] = LM \) as desired, by the definition of a limit.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limit Definition
When we talk about the limit of a function as it approaches a certain value, we are essentially discussing what happens to the values of the function as we get closer and closer to a particular point. A limit exists when the function values approach a specific value as the input approaches a certain point. In mathematical terms, for a function \(f(x)\), we say \(\lim_{x \to a} f(x) = L\), meaning as \(x\) gets closer to \(a\), \(f(x)\) gets closer to \(L\). This concept is fundamental in calculus, laying the groundwork for understanding continuity and derivative. Understanding this definition is crucial for analyzing changes in functions and predicting behaviors near points of interest. This definition also provides the basis for more advanced concepts, including derivatives and integrals.
Limits of Products
The limit of a product deals with functions where you have two separate components multiplied together, like \(f(x)\) and \(g(x)\). To find the limit of such products, we rely on the limit law which states that \(\lim_{x \to a} [f(x) g(x)] = (\lim_{x \to a} f(x))(\lim_{x \to a} g(x)) = LM\) if both limits individually exist. This simplifies the process of finding limits because we can evaluate the limit of each function separately, then multiply them together. It鈥檚 essential because it provides a systematic way to handle limits of more complex functions built from simpler ones.The rule leverages the properties of limits and the arithmetic operations under the limit environment, allowing for the simplifications that make calculus calculations feasible and pragmatic.
Epsilon-Delta Definition
The epsilon-delta definition is a rigorous way to define a limit. It states that for every \(\epsilon > 0\) (a small positive number), there exists a \(\delta > 0\) such that whenever \(0 < |x-a| < \delta\), \(|f(x) - L| < \epsilon\). This means, as we can make \(f(x)\) as close as we want to \(L\) by choosing \(x\) sufficiently close to \(a\). This formalization allows the precise expression and proof of the limit鈥檚 existence and computation, ensuring that we can always find a "small neighborhood" where the function鈥檚 output is close to the limit.It's a foundational component in calculus, ensuring that truths about functions are backed by precise and incontrovertible arguments, validating the truth of equations and statements using arithmetic logic.
Limit Proofs
Limit proofs are logical arguments that demonstrate why a limit holds true according to its definition. They involve showing, step by step, how the assumptions lead to the conclusion that a function approaches a specific value. In the scenario of verifying \(\lim_{x \to a} [f(x)g(x)] = LM\), a systematic approach is taken:
  • Firstly, the epsilon-delta definition helps set boundaries for how small \(f(x) - L\) and \(g(x) - M\) need to be.
  • Next, we leverage algebraic manipulation and properties of inequalities to show that the product function \(f(x)g(x)\) closely surrounds \(LM\).
  • Finally, conclude by demonstrating that the conditions are satisfied, effectively proving the limit exists using the definition.
This detailed progression ensures that the limit holds under mathematical scrutiny, reinforcing both understanding and applicability of limits in various mathematical contexts.

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Most popular questions from this chapter

[T] In the following exercises, use a calculator to draw the graph of each piecewise-defined function and study the graph to evaluate the given limits. \(g(x)=\left\\{\begin{array}{ll}{x^{3}-1,} & {x \leq 0} \\ {1,} & {x>0}\end{array}\right.\) $$a. \lim _{x \rightarrow 0^{-}} g(x)$ $$b. \lim _{x \rightarrow 0^{+}} g(x)$$

In the following exercises, write the appropriate \(\varepsilon-\delta\) definition for each of the given statements. $$\lim _{x \rightarrow c} h(x)=L$$

A particle moving along a line has a displacement according to the function \(x(t)=t^{2}-2 t+4,\) where \(x\) is measured in meters and \(t\) is measured in seconds. Find the average velocity over the time period \(t=[0,2]\)

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In the following exercises, sketch the graph of a function with the given properties. $$\lim _{x \rightarrow-\infty} f(x)=0, \quad \lim _{x \rightarrow-1^{-}} f(x)=-\infty$$ $$\lim _{x \rightarrow-1^{+}} f(x)=\infty, \lim _{x \rightarrow 0} f(x)=f(0), f(0)=1, \lim _{x \rightarrow \infty} f(x)=-\infty$$
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