Problem 36 - APEX Calculus

Question

Use the Squeeze Theorem, where appropriate, to evaluate the given limit. $$ \lim _{x \rightarrow 0} \sin x \cos \left(\frac{1}{x^{2}}\right) $$

Step-by-Step Solution

Answer

The limit is 0 by using the Squeeze Theorem.

Step 1: Recognize the Function Structure

The function given is \( \sin x \cos \left(\frac{1}{x^2}\right) \). Notice that as \( x \rightarrow 0 \), the term \( \cos \left(\frac{1}{x^2}\right) \) involves a very high frequency oscillation, while \( \sin x \) is a smooth function approaching zero.

Step 2: Identify Bounds for the Oscillating Part

The cosine function always satisfies \( -1 \leq \cos \theta \leq 1 \) for any \( \theta \). Thus, for any value \( x \), \( -1 \leq \cos \left(\frac{1}{x^2}\right) \leq 1 \).

Step 3: Construct Inequality Using Squeeze Theorem

Using the bounds from the cosine function, the given function satisfies \(-|\sin x| \leq \sin x \cos \left(\frac{1}{x^2}\right) \leq |\sin x|\).

Step 4: Determine Limit of Bounding Functions

Since \( \sin x \rightarrow 0 \) as \( x \rightarrow 0 \), both \(-|\sin x|\) and \(|\sin x|\) also approach zero. Therefore, both bounds converge to 0.

Step 5: Apply the Squeeze Theorem

The Squeeze Theorem states that if \( a(x) \leq f(x) \leq b(x) \) and \( \lim_{x\to a}a(x) = \lim_{x\to a}b(x) = L \), then \( \lim_{x\to a}f(x) = L \). Hence, \( \lim _{x \rightarrow 0} \sin x \cos \left(\frac{1}{x^2}\right) = 0 \).

Key Concepts

Understanding Limits

When we talk about limits in calculus, we are essentially discussing what happens to a function as the input approaches a particular value. Limits help us analyze the behavior of functions without strictly being at a particular point. They are particularly important when dealing with functions that have indeterminate forms at a point, such as when both the numerator and denominator are zero or when the function appears undefined.

In the original exercise, we are asked to evaluate the limit of the function \( \sin x \cos \left(\frac{1}{x^2}\right) \) as \( x \to 0 \). The term \( \sin x \) approaches zero as \( x \to 0 \), which is straightforward. However, \( \cos \left(\frac{1}{x^2}\right) \) oscillates very rapidly as \( x \) becomes very small, presenting a unique challenge.

The use of the Squeeze Theorem here allows us to evaluate this problematic limit by using simpler limits as reference points or bounds, thus making limits a powerful tool in calculus.

Role of Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus due to their periodic nature, relationships with angles, and intrinsic bounds. These functions commonly become part of limit problems, especially when dealing with oscillating or periodic behaviors.

Let's break down the two trigonometric functions in the exercise:

• \( \sin x \) is smooth and continuous, and it specifically approaches 0 as \( x \to 0 \). Its straightforward behavior makes it easier to handle in limits.
• \( \cos \left(\frac{1}{x^2}\right) \) is more complex because the argument \( \frac{1}{x^2} \) leads to rapid oscillation as \( x \) nears 0. Despite this, cosine itself is restricted to always be between -1 and 1, providing useful bounds for applying the Squeeze Theorem.

Together, these properties of sine and cosine help construct the inequalities necessary for employing the Squeeze Theorem, showcasing how trigonometric functions can be both challenging and helpful in calculus.

Oscillating Functions and Their Characteristics

Oscillating functions are those that fluctuate between values. They repeat certain patterns, generally described by functions like sine and cosine. These oscillations can challenge limits because they might not approach a single value, appearing to lack convergence.

In our specific problem, the term \( \cos \left(\frac{1}{x^2}\right) \) exhibits oscillating behavior that becomes pronounced as \( x \to 0 \). Due to the rapidly changing frequency, it does not settle at a single value. However, understanding its fundamental bounds \(-1 \leq \cos \theta \leq 1\) provides a control mechanism.

By constructing limits on such oscillating components, the Squeeze Theorem shines as a powerful tool to help pinpoint the actual limit value of the larger function. This use in oscillating functions illustrates the important principle that while individual components may not converge, the overall behavior can still be predictable under particular contexts.