Question

Let \(f\) be a nonnegative continuous function defined on \(0 \leq x<\infty\) with \(\int_{0}^{\infty} f(x) d x<\infty .\) Show that (a) if \(\lim _{x \rightarrow \infty} f(x)\) exists it must be 0 ; (b) it is possible that \(\lim _{x \rightarrow \infty} f(x)\) does not exist.

Short answer

(a) If \(\lim_{x \to \infty} f(x)\) exists, it must be 0. (b) It's possible \(\lim_{x \to \infty} f(x)\) does not exist.

Step-by-step solution

Understanding the Given Conditions

We are given a nonnegative continuous function \(f(x)\) on \(0 \leq x < \infty\) and \(\int_0^{\infty} f(x) \, dx < \infty\). This means that the area under the curve of \(f(x)\) from 0 to infinity is finite.

Proving Part (a)

To show that \(\lim_{x \to \infty} f(x) = 0\), if it exists, assume by contradiction that \(\lim_{x \to \infty} f(x) = L > 0\). Then there exists \(M\) such that for all \(x > M\), \(f(x) > \frac{L}{2}\). The integral \(\int_M^{\infty} f(x) \, dx\) would then diverge as it is greater than \(\int_M^{\infty} \frac{L}{2} \, dx\), which is infinite. This contradicts the given condition that \(\int_0^{\infty} f(x) \, dx < \infty\). Thus, \(L\) must be 0.

Exploring Part (b)

For part (b), we construct an example to show \(\lim_{x \to \infty} f(x)\) does not exist. Consider \(f(x) = \sin(x^2) + 1\), which is nonnegative and oscillates between 0 and 2. The integral \(\int_0^{\infty} \sin(x^2) + 1 \, dx\) converges due to the oscillatory nature of \(\sin(x^2)\), but \(\lim_{x \to \infty} f(x)\) does not exist due to its oscillation.

Key concepts

Nonnegative Continuous Functions

Nonnegative continuous functions are a vital concept when dealing with improper integrals. These functions, which are always greater than or equal to zero, play a crucial role in ensuring the area under the curve is defined as "non-negative." A nonnegative continuous function is one which never dips below the x-axis on the graph. This means any integral from 0 to infinity will be either zero or stretch up to a finite, nonnegative value.
Understanding that a function is continuous is all about smoothness. No breaks or abrupt changes should exist in the graph of the function over the domain. Continuous functions that are also nonnegative create a landscape where, even as x approaches infinity, no sudden breaks disrupt the value computation of the integrals.

• Ensures positive area under the curve
• Guarantees no breaks in the domain
• Facilitates calculation of improper integrals

Together, these properties enable us to discuss the convergence of an integral over an infinite domain, which is central in evaluating limits at infinity.

Limit at Infinity

Exploring the limit of a function as x approaches infinity is like peering into a mathematical horizon. When working with improper integrals, particularly with nonnegative continuous functions, determining if and what a function approaches can reveal much about its behavior. The question of whether a function "settles" into a particular value or trends off to infinite values is described by this limit.
If the limit of a function exists, it must settle to zero, especially when the integral corresponding to it is finite. This behavior can be reasoned with a contradiction. If the limit was not zero, the area (integral) from some point to infinity would itself be infinite, which opposes our initial assumption of a finite integral. Therefore, we logically conclude that the function dwindles to a near-zero outside value.

• Helps verify function convergence
• Ensures the function doesn't diverge to infinity
• Directly impacts whether the related integral converges

The concept can seem abstract, but it's an essential tool for validating the behaviors of various functions over an infinite span.

Oscillatory Functions

Oscillatory functions introduce an intriguing twist within the realm of limits and integrals. Unlike simply converging functions, oscillatory functions fluctuate between values, which can sometimes cause the limit at infinity to not exist. For instance, a function like \(f(x) = \sin(x^2) + 1\) never settles into one single value because the \(\sin(x^2)\) component keeps alternating between predefined limits.
This constant fluctuation results in a function that doesn't stabilize as x approaches infinity. The resulting behavior is complex because, despite not having a definitive limit, such functions can still have finite integrals. Thanks to their oscillatory behavior, the overall "area" under the curve doesn't infinitely accumulate, which keeps the integral from diverging.

• Describes functions with recurrent ups and downs
• Explains why some functions lack a limit at infinity
• Still allows for a convergent integral under certain conditions

Despite appearing unpredictable on first glance, oscillatory functions play a perfectly rhythmic role in the complex harmonics of mathematical integration.