Problem 18 - Precalculus: Mathematics for Calculus

Question

Find the limit. $$\lim _{x \rightarrow \infty} \sin ^{2} x$$

Step-by-Step Solution

Answer

The limit does not exist since \(\sin^2 x\) oscillates between 0 and 1 indefinitely.

Step 1: Understand the Problem

The expression given involves finding the limit as \(x\) approaches infinity of \(\sin^2 x\). The sine function, \(\sin x\), oscillates between -1 and 1 for all \(x\). Thus, \(\sin^2 x\) will oscillate between 0 and 1.

Step 2: Consider the Range of the Squared Sine Function

Since \(\sin x\) takes values between -1 and 1, squaring \(\sin x\) results in \(\sin^2 x\) taking values between 0 and 1. Therefore, \(0 \leq \sin^2 x \leq 1\).

Step 3: Discuss the Oscillating Nature

\(\sin x\) oscillates endlessly as \(x\) increases, meaning \(\sin^2 x\) will also indefinitely oscillate between 0 and 1 without settling to a single value. This oscillation implies that the limit does not exist since it does not approach any specific finite value as \(x\) approaches infinity.

Key Concepts

Trigonometric Functions

Trigonometric functions like sine and cosine are fundamental in mathematics, particularly in understanding periodic phenomena. These functions repeat their values in a predictable manner, which is known as "oscillation." Specifically, the function \( \sin x \) varies smoothly between -1 and 1 as the angle \( x \) increases. This periodic behavior is denoted by the function's wave-like pattern, which repeats every \( 2\pi \) radians. If we take \( \sin x \) and square it—resulting in \( \sin^2 x \)—the negative values disappear, and we obtain values in the range of 0 to 1. This is because squaring any real number always yields a non-negative result. Understanding these fundamental characteristics of trigonometric functions is essential when dealing with limits relating to them.

Infinite Limits

In the context of limits, "infinite" describes the behavior of a function as the input grows larger and larger without bound. Limits could result in a defined number, approach infinity, or not exist at all. For functions like \( \sin^2 x \), which are rooted in oscillating behavior, we are particularly interested in how they behave as \( x \to \infty \). Unlike other functions where we observe a trend, \( \sin^2 x \) does not converge towards any particular number. Historically, mathematicians have explored limits to understand and quantify behaviors of functions in extreme scenarios, and infinite limits help illustrate how functions might "act" at their extremes.

Oscillating Functions

An oscillating function is one that continues to cycle through a range of values without settling at a particular point, much like \( \sin x \). With \( \sin^2 x \), this oscillation is also evident, but constrained between 0 and 1. As \( x \) approaches infinity, the function does not settle; it persists in this waving motion without convergence. Such behavior results in a limit that does not exist because the function continually "bounces" between values instead of approaching a specific number. Mathematically, if a function continually flits between numbers without trending toward a singular value, it causes the limit to be undeclared or nonexistent.