Problem 60 - Calculus 3

Question

For the following exercises, find the limit of the function. $$ \lim _{(x, y) \rightarrow(1,2)} x $$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Understand the Limit Definition

In this problem, we need to find the limit of the function as the points \((x, y)\) approach \((1,2)\). The function given is simply \(f(x, y) = x\). This means we want to know what value \(x\) approaches as \(x\) gets close to 1 and \(y\) gets close to 2.

Step 2: Focus on the Relevant Variable

The important aspect of this limit problem is that the function only depends on \(x\). The variable \(y\) does not influence the function \(f(x, y) = x\). Thus, the function's value is solely determined by \(x\). As \(x\) approaches 1, \(f(x, y)\) approaches \(1\) as well, regardless of the value of \(y\).

Step 3: Conclude the Limit

Since \(f(x, y) = x\) doesn't depend on \(y\), the limit is the value that \(x\) approaches as \(x\) gets close to 1. Therefore, the limit of \(f(x, y)\) as \((x, y)\) approaches \((1, 2)\) is simply \(1\).

Key Concepts

Multivariable Calculus

Multivariable calculus is an extension of calculus that involves functions of several variables. It builds upon the ideas of single-variable calculus by incorporating more complex functions with two or more independent variables, such as \((x, y)\).
In multivariable calculus, we can explore

• partial derivatives, which measure the rate of change of a function with respect to one variable while keeping others constant;
• multiple integrals, allowing us to calculate volumes under surfaces or areas bounded by curves;
• limits of functions, which help us understand how functions behave as they approach certain points in a multidimensional space.

The study of limits in multivariable calculus can be more complicated than in single-variable calculus because there are multiple paths to approach a given point. This makes understanding the concept of limits crucial for analyzing and predicting the behavior of multivariable functions.

Approaching a Point

In multivariable calculus, approaching a point involves considering the behavior of a function as the variables get close to a specific set of values. For example, in our exercise, we examine what happens as \((x, y)\) approaches \((1, 2)\).
This poses a unique challenge compared to single-variable calculus, where only one path is considered along the number line. Here,

• we might approach from different angles, such as diagonally or directly along one of the axes;
• any approach must consistently lead to the same value for a limit to exist at that point. If different paths yield different limit values, the overall limit does not exist.

Approaching \((1, 2)\) means that the values of \(x\) and \(y\), as independent variables, get closer to these points simultaneously from any possible direction. In our specific problem, however, the function depends only on \(x\), making the exploration along variable \(y\) irrelevant to the outcome of the limit.

Function Dependency

Function dependency is a key concept in understanding how a function behaves as its variables change. Typically, in functions with more than one variable, each variable can alter the function's output.
In the case of the exercise, we analyze the function \(f(x, y) = x\), which demonstrates a unique type of dependency because:

• the function is solely dependent on the variable \(x\), meaning changes in \(y\) have no effect on the function's value;
• This simplification can make solving limits easier as it reduces the problem from a multivariable to a single-variable scenario.

Understanding the dependency clarifies why the limit is \(1\), based solely on \(x\)'s approach to \(1\). In more complex functions, checking all variables' influence is crucial to determine how they collectively affect the limit.