91Ó°ÊÓ

The concept of the limit of a function is crucial in calculus, especially when analyzing the behavior of functions as they approach a particular point. A limit helps us understand the value that a function tends towards as the input gets closer to some point, even if the function is not explicitly defined at that point.
In this context, the Sandwich Theorem is instrumental. This theorem enables us to find the limit of a function that is "sandwiched" between two other functions whose limits are known. If a function \(f(x, y)\) is bounded by two other functions \(g(x, y)\) and \(h(x, y)\) such that \(g(x, y) \leq f(x, y) \leq h(x, y)\) and both \(g(x, y)\) and \(h(x, y)\) approach the same limit \(L\) as \((x, y)\) approaches \((x_0, y_0)\), then \(f(x, y)\) must also approach \(L\).
In essence, limits help describe the behavior of functions at points that may be difficult or impossible to evaluate directly.

Functions of two variables are expressions that depend on two independent factors, typically denoted as \(x\) and \(y\). Such functions, written as \(f(x, y)\), can describe phenomena that change based on two inputs, unlike single-variable functions.
These functions are evaluated in a plane, and the insights from studying them extend our understanding of how changes in one variable impact the function concerning another. When dealing with the limit of functions of two variables, the point of interest, like \((x_0, y_0)\), is approached from any direction within the plane.
Considering our exercise, we examined the limit \(\lim_{(x, y) \rightarrow (0,0)} x \cos \frac{1}{y}\). Here, \(x\) and \(y\) are two independent variables. The goal is to analyze how the function behaves as \(x\) and \(y\) simultaneously close in on zero, providing deeper understanding into multi-variable calculus.

Inequalities are essential tools in calculus, often used to estimate or restrict the values a function can take. In this problem, we use inequalities to bound \(f(x, y) = x \cos \frac{1}{y}\) between two simple functions.
Given that \(|\cos(1/y)| \leq 1\), multiplying through by \(x\) yields \(-x \leq x \cos(1/y) \leq x\). This inequality is a pivotal step in applying the Sandwich Theorem. By controlling the boundaries within which \(f(x, y)\) can vary, we ensure that when both bounds tend to zero, the target function \(f(x, y)\) also conforms to this limit.
Inequalities thus facilitate the application of powerful theorems and ensure accurate evaluations of limits, allowing us to deduce consistent conclusions from otherwise complex problems.