Problem 108 - Calculus Volume 3

Question

Use polar coordinates to find \(\lim _{(x, y) \rightarrow(0,0)} \cos \left(x^{2}+y^{2}\right)\).

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Convert to Polar Coordinates

In polar coordinates, we express \(x\) and \(y\) in terms of \(r\) and \(\theta\) as: \(x = r \cos \theta\) and \(y = r \sin \theta\). Therefore, \(x^2 + y^2\) becomes \(r^2\). Substitute to get the expression: \(\cos(x^2 + y^2) = \cos(r^2)\).

Step 2: Express the Limit in Terms of r

We need to evaluate the limit \(\lim_{(x, y) \to (0,0)} \cos(x^2 + y^2)\), which becomes \(\lim_{r \to 0} \cos(r^2)\) when using polar coordinates.

Step 3: Evaluate the Limit

As \(r\) approaches 0, the expression \(r^2\) also approaches 0. Therefore, we need to find \(\cos(0)\). Since \(\cos(0) = 1\), it follows that: \(\lim_{r \to 0} \cos(r^2) = 1\).

Key Concepts

Limit Evaluation

Evaluating limits might seem tricky at first, but they are a fundamental aspect of calculus, providing insight into the behavior of functions as they approach certain points. In multivariable calculus, you often evaluate limits as two or more variables approach a particular point. One common point is the origin, \(x \, \to \, 0, \, y \, \to \, 0\). In our exercise, we are tasked with evaluating the limit of \(\cos(x^2 + y^2)\) as \(x\) and \(y\) both approach zero.

By analyzing the approach of these variables, we effectively determine the behavior of the function near that point. Evaluating limits helps us describe and understand the continuity and potential discontinuities or unusual behavior of functions. Ultimately, limit evaluation is crucial for analyzing and understanding more complex mathematical systems, aiding in the transition from understanding elementary algebra to grasping advanced calculus concepts.

Conversion to Polar Coordinates

To simplify complex problems in multivariable calculus, converting Cartesian coordinates (x, y) to polar coordinates (r, \(\theta\)) can be highly effective. This conversion replaces the typical x and y with r and the angle \(\theta\), where \(x = r \cos \theta\) and \(y = r \sin \theta\). Importantly, \(x^2 + y^2\) converts directly to \(r^2\) in polar coordinates.

This transformation can greatly simplify limit evaluation, especially when dealing with circular symmetry around the origin. Applying it to our limit problem transforms the cumbersome \(\cos(x^2 + y^2)\) into the much neater form of \(\cos(r^2)\). This makes the evaluation as \(r \, \to \, 0\) easier, and confirms that polar coordinates are fantastic for dealing with radial limits or any situations involving circular or rotational systems. Polar conversion often leads to a clearer perspective on the problem at hand as it allows the simplification of complex algebraic expressions.

Trigonometric Limits

Trigonometric limits are a key aspect of understanding the behavior of trigonometric functions at particular points, often where direct substitution might be undefined or lead to indeterminate forms. In our particular problem, \(\cos(x^2 + y^2)\) changes into \(\cos(r^2)\) in polar coordinates.

The key property of the cosine function in trigonometry that makes solving this problem straightforward is its continuity. As \(r^2\) approaches zero, by the property of continuity, \(\cos(r^2)\) approaches \(\cos(0)\), which simplifies directly to 1. This property of \(\cos\) aids greatly when handling trigonometric limits in calculus.

Understanding these limits allows mathematicians and students alike to predict the behavior of trigonometric expressions within functions and systems, providing a foundation for dealing with more intricate mathematical challenges.