91Ó°ÊÓ

Limit evaluation is a fundamental concept in calculus that helps us understand the behavior of functions as their input values approach a certain point. In this exercise, we have two functions, \( f(x) = x^2 \cos\left(\frac{1}{x}\right) \) and \( g(x) = x \), and we are tasked with finding the limit \( \lim_{{x \to 0}} \frac{{f(x)}}{{g(x)}} \).

First, we simplify this expression to \( \lim_{{x \to 0}} x \cos\left(\frac{1}{x}\right) \). Now, considering that the cosine function is always bounded between \(-1\) and \(1\), the entire expression is bounded by \(-x\) and \(x\).

As \( x \) approaches \( 0 \), both \(-x\) and \(x\) also approach \( 0 \). By applying the Squeeze Theorem, which states that if a function is trapped between two other functions that both converge to the same limit, then it also converges to that limit, we find that \( \lim_{{x \to 0}} x \cos\left(\frac{1}{x}\right) \) is \( 0 \). Hence, \( L = 0 \).

Recognizing this approach allows us to evaluate the limits of other expressions where direct substitution isn't possible, making it a powerful tool in calculus.

The Squeeze Theorem is an essential method for finding limits of functions that are difficult to handle directly. It's particularly useful when dealing with functions that involve oscillatory behavior, as seen in this exercise with \( \cos\left(\frac{1}{x}\right) \).

In the context of our problem, the Squeeze Theorem supplies a crucial strategy. When given a function \( f(x) \), if two simpler functions \( a(x) \) and \( b(x) \) can be found such that \( a(x) \leq f(x) \leq b(x) \) for all \( x \) in some interval containing the point where \( x \) approaches zero (excluding possibly \( x = 0 \) itself), and \( \lim_{{x \to c}} a(x) = \lim_{{x \to c}} b(x) = L \), then \( \lim_{{x \to c}} f(x) = L \) as well.

For our function \( x \cos\left(\frac{1}{x}\right) \), knowing \(-1 \leq \cos\left(\frac{1}{x}\right) \leq 1\) allows us to squeeze our function between \(-x\) and \(x\).

Since both limits of \(-x\) and \(x\) as \( x \to 0 \) are zero, the Squeeze Theorem assures us that the limit of \( x \cos\left(\frac{1}{x}\right) \) as \( x \to 0 \) is likewise zero. It's an effective technique in calculus for pinning down limits within bounding values.

Oscillatory limits occur in calculus when the function consistently changes between values instead of approaching a single value. This is perfectly illustrated in our exercise.

In part (b) of the problem, we need to evaluate \( \frac{{f'(x)}}{{g'(x)}} = \sin\left(\frac{1}{x}\right) - \frac{1}{x} \cos\left(\frac{1}{x}\right) \) as \( x \to 0 \).

Firstly, the sine function, \( \sin\left(\frac{1}{x}\right) \), oscillates between -1 and 1 without settling. This behavior creates what's known as an oscillatory limit, meaning it doesn't converge to a single number as \( x \to 0 \). Similarly, \( \frac{1}{x} \cos\left(\frac{1}{x}\right) \) shows complex oscillations as \( x \) approaches \( 0 \), since both its multiplicative factors contribute to its unpredictable and unbounded behavior.

When these oscillatory behaviors are part of a ratio in limit calculations, as seen in our expression, it prevents the limit from existing. Simply put, no single number is approached as \( x \) draws nearer to zero. This scenario reflects why the limit \( \frac{{f'(x)}}{{g'(x)}} \) doesn't exist in this context.

Derivative calculations play a pivotal role in calculus. They allow us to understand the rate at which a function changes. In this exercise, derivatives determine the nature of the function's limit as \( x \to 0 \).

We've seen that \( f(x) = x^2 \cos\left(\frac{1}{x}\right) \) results in \( f'(x) = \sin\left(\frac{1}{x}\right) + 2x \cos\left(\frac{1}{x}\right) \). Meanwhile, \( g'(x) = 1 \) because \( g(x) = x \), which is linear.

Calculating derivatives of functions helps us identify the behavior of functions as they approach particular values. In the given expressions, notice that \( \sin\left(\frac{1}{x}\right) \), arising from the chain rule while differentiating \( x^2 \cos\left(\frac{1}{x}\right) \), reflects an interesting point: crescent oscillations affect the limits at zero, causing a lack of convergence.

Hence, these derivative calculations reveal critical insights: despite \( \frac{{f(x)}}{{g(x)}} \to 0 \) (because \( f(x) \)'s highest-order derivatives affect limits differently), \( \frac{{f'(x)}}{{g'(x)}} \) behaves erratically due to the oscillating nature of \( \sin\left(\frac{1}{x}\right) \) and the complexities inherent in their ratio as \( x \to 0 \).

This example shows how derivatives can illuminate the intricate behaviors obscured by standard function limits, emphasizing the importance of accurate derivative calculations in understanding different limits.