Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the indicated limit or state that it does not exist. \(\lim _{(x, y) \rightarrow(0,0)} \frac{\sin \left(x^{2}+y^{2}\right)}{x^{2}+y^{2}}\)

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Identify the Form

We recognize that the limit is in the form \( \frac{\sin(a)}{a} \) as \( a \to 0 \). This is a standard limit in calculus that equals \(1\). In this problem, \( a = x^2 + y^2 \).

Step 2: Substitute and Simplify

Consider \( a = x^2 + y^2 \). As \((x, y) \to (0,0)\), it implies that \( a \to 0 \). The expression becomes \( \lim_{a \to 0} \frac{\sin(a)}{a} \).

Step 3: Apply the Standard Limit

Using the known limit \( \lim_{a \to 0} \frac{\sin(a)}{a} = 1 \), we can directly apply this to conclude that the given limit also equals 1.

Key Concepts

Multivariable Calculus

Multivariable calculus is an extension of calculus that deals with functions of multiple variables. Instead of working with a single variable, such as in traditional calculus, we work with functions involving two or more variables. This allows us to explore complex systems in higher dimensions.

One of the main interests in multivariable calculus is analyzing how these functions change and behave as their inputs change. The tools and techniques developed for this purpose include partial derivatives, gradients, and most importantly for our topic, limits.

Limits in multivariable calculus help us understand how a function behaves near a point by examining the trend of its values as the inputs approach that point. These calculations can become more complex, as we have to consider paths along which these inputs approach the point, unlike single-variable calculus where there's only one path.

Limits in Two Variables

Limits in two variables require us to analyze functions as both variables approach a particular point. In contrast to single-variable limits, where there is typically a single pathway, two-variable limits have countless paths that could be taken to approach a point.

This can make determining limits complex, as the function may behave differently depending on the path taken. Therefore, a limit only exists in two variables if all possible paths result in the same value.

In our example, \[ \\lim_{(x, y) \rightarrow(0,0)} \frac{\sin (x^{2}+y^{2})}{x^{2}+y^{2}} \\]we notice that the form resembles a well-known limit situation in single-variable calculus. Recognizing and correctly formulating the problem allows us to simplify greatly and determine if the limit exists, and what it might be.

Standard Limits in Calculus

Standard limits in calculus provide us with essential rules and results that make solving limits more straightforward. One of these is the limit: \[ \lim_{a \to 0} \frac{\sin(a)}{a} = 1 \]This is a foundational result often used to solve more complex problems.

In the context of multivariable calculus, figuring out if we can transform a given problem into one that uses a standard limit such as this can save significant time and effort. For example, in our problem, we identified that as \((x^2 + y^2) \to 0\), we are essentially considering \(\frac{\sin(a)}{a}\) as a approaches 0.

This key insight is why recognizing and applying standard limits is crucial. They allow us to simplify multivariable limit problems and come to the correct conclusion quickly and confidently.