Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f\) is continuous on \([0, \infty)\) and \(\int_{0}^{\infty} f(x) d x\) diverges, then \(\lim _{x \rightarrow \infty} f(x) \neq 0\)

Short answer

The statement is False. A counterexample is the function \(f(x) = 1/x\), which is continuous on \([0, \infty)\), its integral from \(0\) to infinity diverges, and its limit as \(x\) goes to infinity is zero.

Step-by-step solution

Verifying the Statement

Start with the assumption that the given statement is true. That is, if the integral of the function \(f\) from \(0\) to infinity diverges, then the limit of \(f(x)\) as \(x\) approaches infinity is not equal to \(0\).

Counterexample Search

Attempt to find a counterexample to the statement. A simple function that is continuous for all \(x \geq 0\) and whose integral from \(0\) to infinity is divergent, where limit as \(x\) approaches infinity is \(0\), would contradict the original statement.

Counterexample Definition

Define such a function. A suitable counterexample here is \(f(x) = 1/x\). It is continuous on \((0, \infty)\), the integral \(\int_{1}^{\infty} 1/x dx = \ln|x| |_{1}^{\infty}\) diverges, but the limit as \(x\) approaches infinity, \(\lim _{x \rightarrow \infty} 1/x = 0\). The original statement is thereby contradicted and thus is False.

Statement Conclusion

Conclude that the statement: 'If a function \(f\) is continuous on \([0, \infty)\) and the integral from \(0\) to infinity of \(f\) diverges, then the limit of \(f(x)\) as \(x\) approaches infinity is not zero' is False.

Justification

Justify the conclusion. This is because a counterexample exists: \(f(x) = 1/x\), which is continuous on \([0, \infty)\), its integral diverges, and its limit as \(x\) goes to infinity is zero. Therefore, the integrability of a function over an interval and the limit of the function at the bounds of that interval do not have the direct relationship suggested by the statement.

Key concepts

Counterexamples in Calculus

Counterexamples play a crucial role in understanding the boundaries and limitations of mathematical statements. They serve as evidence that a general statement does not hold in all cases. For instance, the statement, 'If the integral of a function from 0 to infinity diverges, then the limit of the function as it approaches infinity is not zero,' can be disproven with a counterexample.

Consider the function f(x) = 1/x. It is continuous on (0, \(\infty\)). However, despite being continuous, the integral \(\int_{1}^{\infty} 1/x \, dx\) diverges. Meanwhile, the limit of f(x) as x approaches infinity is indeed zero: \(\lim _{x \rightarrow \infty} 1/x = 0\). This specific example shows that the integral's divergence does not necessitate that the function's limit is different from zero, thereby invalidating the statement through a counterexample.

Properties of Continuous Functions

Continuous functions have properties that are both useful and intuitive, allowing for a smooth graph with no breaks or holes. A function f is said to be continuous on an interval if for every point x within the interval, the limit of f(x) as it approaches that point is equal to the function's value at that point: \(\lim_{x \rightarrow c} f(x) = f(c)\). Moreover, continuous functions on closed intervals have some noteworthy properties, such as the ability to attain maximum and minimum values on those intervals.

However, the behavior of continuous functions over an infinite interval, such as [0, \(\infty\)), can be more complex. For example, a function can be continuous over such an interval and still have an improper integral that diverges, which does not imply a particular behavior for the function as x approaches infinity.

Improper Integrals

Improper integrals are used when evaluating the integral of a function over an unbounded interval or when the function has an infinite discontinuity. An integral is considered improper if either the limits of integration include infinity, such as \(\int_{a}^{\infty} f(x) \, dx\), or if the integrand becomes infinite at some point within the interval of integration.

A crucial aspect in the evaluation of improper integrals is the concept of convergence or divergence. If the limit exists and is finite as the interval expands to infinity or approaches a point of discontinuity, the integral is said to 'converge.' If the limit does not exist or is infinite, the integral 'diverges.' The interesting interplay between the convergence of improper integrals and the behavior of their respective functions at infinity is a central topic in calculus.

Limits at Infinity

Limits at infinity investigate the behavior of functions as the input grows without bound. Specifically, they describe what value a function approaches as the independent variable x increases or decreases without limit. Symbolically, this is denoted as \(\lim_{x \rightarrow \infty} f(x)\) and \(\lim_{x \rightarrow -\infty} f(x)\), respectively.

The value a function approaches may be a finite number, it might be infinite, or the limit might not exist at all. A limit at infinity being zero, as seen in the counterexample \(f(x) = 1/x\), does not necessarily determine the integrability of a function or guarantee that an integral over an infinite interval will converge, which is a common misconception.