Problem 44 - Calculus

Question

Is the statement true or false? Assume that \(y=f(x)\) is a solution to the equation \(d y / d x=g(x) .\) If the statement is true, explain how you know. If the statement is false, give a counterexample. If \(\lim _{x \rightarrow \infty} g(x)=0,\) then \(\lim _{x \rightarrow \infty} f(x)=0.\)

Step-by-Step Solution

Answer

False. Counterexample: \(g(x) = \frac{1}{x}\) leads to \(f(x) = \ln|x| + C\).

Step 1: Understanding the Given Equation

We are given that the function \(y=f(x)\) is a solution to the differential equation \(\frac{dy}{dx}=g(x)\). This suggests that the derivative of \(f(x)\) with respect to \(x\) is \(g(x)\).

Step 2: Analyze the Given Limit Condition

The statement claims that if \(\lim_{x \to \infty} g(x) = 0\), then \(\lim_{x \to \infty} f(x)\) should also be 0. We're tasked to evaluate this claim.

Step 3: Integrating the Differential Equation

Since \(\frac{dy}{dx}=g(x)\), we integrate both sides to find \(f(x)\). This gives \(f(x) = \int g(x) \, dx + C\), where \(C\) is the constant of integration.

Step 4: Evaluate the Limit of the Integral

Even if \(\lim_{x \to \infty} g(x) = 0\), it does not follow that \(\lim_{x \to \infty} \int g(x) \, dx + C = 0\). An integral of a function approaching zero can still diverge or approach another constant.

Step 5: Providing a Counterexample

Consider the function \(g(x) = \frac{1}{x}\). Here, \(\lim_{x \to \infty} \frac{1}{x} = 0\). However, \(f(x) = \int \frac{1}{x} \, dx = \ln|x| + C\), indicates that \(\lim_{x \to \infty} f(x) = \infty\), not 0.

Step 6: Conclusion: Determining the Truth Value of the Statement

The statement is false. Just because \(g(x)\) approaches zero, it doesn't ensure \(f(x)\) approaches zero, as demonstrated by the counterexample.

Key Concepts

Limits at Infinity

Limits at infinity deal with the behavior of functions as the input, often denoted as variable \(x\), approaches very large positive or negative values. When we talk about \( \lim_{x \to \infty} g(x) \), we are asking what value, if any, \(g(x)\) gets closer to as \(x\) becomes infinitely large. In the context of differential equations, knowing the limit of \(g(x)\) can tell us about the long-term behavior of its corresponding integral function \(f(x)\). However, it’s crucial to understand that just because \(g(x)\) approaches zero, doesn't mean \(f(x)\) will also approach zero. To think about limits in a practical sense:

• Imagine driving your car towards a distant city (the point at infinity). \(g(x)\) represents your speed. Even if you slow down and eventually stop (as \(g(x) \to 0\)), your total distance traveled (\(f(x)\)) could continue to increase and vary.
• This difference between speed and distance is analogous to the difference between a function and its integrated value.

Remember, the value \(g(x)\) approaches isn't always indicative of the behavior of \(f(x)\). This nuance is vital in calculus, as seen in more advanced math problems.

Integration

Integration is a fundamental concept in calculus used to find functions when their derivatives are known. Essentially, when integrating a function like \(g(x)\) to find \(f(x)\), we determine an "accumulated" total. This is often represented as \(f(x) = \int g(x) \, dx + C\), where \(C\) is the constant of integration resulting from the indefinite integral.Think of integration as the opposite process of differentiation.

• Differentiation figures out the rate of change (the slope of a function).
• Integration accumulates the total impact of these changes over an interval.

In our differential equation \(\frac{dy}{dx} = g(x)\), we integrate \(g(x)\) to find the solution \(f(x)\). Despite \(g(x)\) becoming zero at infinity, the integral may still diverge dramatically based on its overall behavior. An example is \(g(x) = \frac{1}{x}\). It tends to zero as \(x\) approaches infinity. However, integrating it yields \(f(x) = \ln|x| + C\), which does not flatten out but actually diverges without bound.

Counterexamples

Counterexamples are incredibly useful in mathematics, acting as critical tools to disprove statements. A counterexample provides an instance where the hypothesis of a statement holds true, but the subsequent conclusion fails. This single instance invalidates the assumption of a universally true statement.In the given exercise, the statement claims if \(\lim_{x \to \infty} g(x) = 0\), then \(\lim_{x \to \infty} f(x) = 0\). To disprove this, we employ a counterexample:

• Consider \(g(x) = \frac{1}{x}\). Clearly, \(\lim_{x \to \infty} \frac{1}{x} = 0\).
• Upon integration, \(f(x) = \ln|x| + C\), where \(C\) is the constant of integration.
• As \(x\) approaches infinity, \(\ln|x|\) increases without bound, illustrating \(\lim_{x \to \infty} f(x) = \infty\).

This example conclusively shows that, despite \(g(x)\) approaching zero, \(f(x)\) does not. Thus, the original statement is false. Counterexamples like this highlight the importance of not assuming a direct relationship between a function’s derivative and its limit behavior.