Problem 51 - Calculus with Concepts in Calculus

Question

Suppose there is a number \(M\) such that \(f(x) \geq 0\) for all \(x \geq M\), and assume that \(\lim _{x \rightarrow \infty}[f(x)]^{2}=L .\) Show that \(\lim _{x \rightarrow \infty} f(x)=\sqrt{L}\)

Step-by-Step Solution

Answer

\(\lim_{x \to \infty} f(x) = \sqrt{L}\) due to non-negativity.

Step 1: Understand the Limit Condition

We are given that \( \lim_{x \to \infty} [f(x)]^{2} = L\). This means that as \(x\) approaches infinity, \([f(x)]^2\) approaches \(L\). Since a squared term is given, \(f(x)\) must align with one of its square roots as \(x\) approaches infinity.

Step 2: Apply the Non-negativity Condition

Since \(f(x) \geq 0\) for all \(x \geq M\), this indicates that \(f(x)\) is non-negative for large values of \(x\). Thus, we only consider the non-negative square root of \(L\), which is \(\sqrt{L}\).

Step 3: Deduce the Limit of \(f(x)\)

Given that \([f(x)]^{2} \to L\) and \(f(x) \geq 0\), it follows that \(f(x)\) must be approaching one of its roots, specifically \(\sqrt{L}\). Thus, \(\lim_{x \to \infty} f(x) = \sqrt{L}\), as any other value would make \([f(x)]^{2}\) diverge from \(L\).

Key Concepts

Continuity

When discussing limits and functions, continuity plays a crucial role in understanding the behavior of a function over an interval. In mathematics, a function is said to be continuous if, intuitively speaking, you can draw it without lifting your pencil from the paper. This means there are no breaks, jumps, or holes in the graph of the function.

To be more precise, a function \( f(x) \) is continuous at a point \( x = c \) if the following three conditions are met:

• \( f(c) \) is defined, meaning there is a value at \( x = c \).
• The limit of \( f(x) \) as \( x \to c \) exists.
• The limit of \( f(x) \) as \( x \to c \) is equal to \( f(c) \).

In the context of limits approaching infinity, continuity ensures that the function's value evolves smoothly as \( x \) gets larger and larger. If \( f(x) \) is continuous, the approach to later values, like \( \sqrt{L} \), happens in a predictable manner. This is important in proving limits or deducing one function's behavior based on the limiting behavior of another, like \([f(x)]^2\) in our exercise.

Infinity

When we talk about limits that involve infinity, we're considering the behavior of a function as the variable \( x \) grows without bound. Infinity, symbolized as \( \infty \), is not a number; instead, it's a concept representing unboundedness.

In the exercise, we examine \( \lim_{x \to \infty} [f(x)]^2 = L \). This limit expression tells us what \([f(x)]^2\) approaches as \( x \) increases indefinitely. Understanding limits with infinity involves exploring:

• Convergent behavior: As \( x \) goes to infinity, \( [f(x)]^2 \) approaches a particular number \( L \), suggesting a stabilizing or balancing effect.
• Function behavior at large \( x \): Rather than focusing on small values of \( x \), we direct our attention to the trend of the function as \( x \) increases.
• Asymptotic behavior: This describes how functions behave as they tend towards infinity, helping define horizontal asymptotes or end behavior.

Understanding these concepts helps explain why \( f(x) \) in our problem scenario approaches \( \sqrt{L} \) rather than behaving erratically or unpredictably.

Non-negative Functions

Non-negative functions are those where the function values are never less than zero for a certain domain. In mathematical terms, a function \( f(x) \) is non-negative over a domain \( x \geq M \) if \( f(x) \geq 0 \) for all \( x \) greater than or equal to \( M \).

In our exercise, the non-negative condition is crucial because it simplifies the possibilities of the square root. Since \( f(x) \geq 0 \) for all \( x \geq M \), when \([f(x)]^2\) approaches \( L \), \( f(x) \) must approach something that maintains this non-negative property: \( \sqrt{L} \) (where \( L \) is assumed non-negative because it's the square of a non-negative function).

• If \( L > 0 \), \( \lim_{x \to \infty} f(x) = \sqrt{L} \), smoothly reflecting its non-negative nature.
• If \( L = 0 \), \( \lim_{x \to \infty} f(x) = 0 \), remaining non-negative, with \( f(x) \) tending towards zero.

Non-negativity avoids potential complications with imaginary numbers (which would arise from negatives under square roots) and helps maintain a clear intuitive understanding of how functions behave in this context of limits.