Problem 60
Question
For the following exercises, find the limit of the function. $$\lim _{(x, y) \rightarrow(1,2)} x$$
Step-by-Step Solution
Verified Answer
The limit is 1.
1Step 1: Identify the Function
The function given is simply the variable \( x \). We are tasked with finding the limit of this function as \( (x, y) \rightarrow (1, 2) \).
2Step 2: Understand the Concept of Limit
A limit\( \lim _{(x, y) \rightarrow (a, b)} f(x, y) \) describes the behavior of the function \( f(x, y) \) as the point \((x, y)\) approaches \((a, b)\). In this problem, we are focusing on the variable \( x \) as \( (x, y) \) approaches \((1, 2)\).
3Step 3: Evaluate the Limit
Since the function is \( x \), it is independent of \( y \). Therefore, as \( (x, y) \) approaches \( (1, 2) \), the value of \( x \) approaches \( 1 \). Thus, \( \lim_{(x, y) \rightarrow (1,2)} x = 1 \).
Key Concepts
Limits of FunctionsTwo-Variable FunctionsApproaching Points
Limits of Functions
When dealing with limits in multivariable calculus, we are looking at how a function behaves as the input variables approach a particular point. In simpler terms, it's about understanding how the output of a function becomes close to a particular value as the inputs near specific values.
One key aspect of limits is continuity. A function is continuous at a point if its limit exists at that point and the limit is equal to the function's value there. When calculating limits, we often assume the function is continuous unless stated otherwise. If the functions you are working with are straightforward, like a simple variable or linear function, finding limits becomes much more manageable.
Common techniques to find limits include direct substitution, where we plug in the values that the variables are approaching. Sometimes, especially with complex functions, you might need more advanced techniques like factoring, rationalizing, or employing the Squeeze Theorem. For our specific exercise, where the function is just "x," we see the limit focuses solely on when "x" approaches 1, making the solution straightforward. The variable "y" does not impact the output for this particular limit.
One key aspect of limits is continuity. A function is continuous at a point if its limit exists at that point and the limit is equal to the function's value there. When calculating limits, we often assume the function is continuous unless stated otherwise. If the functions you are working with are straightforward, like a simple variable or linear function, finding limits becomes much more manageable.
Common techniques to find limits include direct substitution, where we plug in the values that the variables are approaching. Sometimes, especially with complex functions, you might need more advanced techniques like factoring, rationalizing, or employing the Squeeze Theorem. For our specific exercise, where the function is just "x," we see the limit focuses solely on when "x" approaches 1, making the solution straightforward. The variable "y" does not impact the output for this particular limit.
Two-Variable Functions
Two-variable functions are functions that take two inputs and produce a single output. These are typically written as \( f(x, y) \), where \( x \) and \( y \) can independently vary. These functions are visualized in 3D space, giving them a surface-like quality that is interesting to study.
Examples of two-variable functions include things like \( f(x, y) = x^2 + y^2 \) or something as complex as \( f(x, y) = \sin(x) \cdot \cos(y) \). These functions can portray more real-life phenomena like temperature maps, altitude charts, and even economic models, where having two variables to adjust can offer richer analysis.
The exercise at hand simplifies our work because it essentially treats "y" as irrelevant for determining the limit. This is because the expression of our function doesn't depend on "y" at all. However, it's crucial when dealing with other more intricate functions to consider how each input distinctly affects the output.
Examples of two-variable functions include things like \( f(x, y) = x^2 + y^2 \) or something as complex as \( f(x, y) = \sin(x) \cdot \cos(y) \). These functions can portray more real-life phenomena like temperature maps, altitude charts, and even economic models, where having two variables to adjust can offer richer analysis.
The exercise at hand simplifies our work because it essentially treats "y" as irrelevant for determining the limit. This is because the expression of our function doesn't depend on "y" at all. However, it's crucial when dealing with other more intricate functions to consider how each input distinctly affects the output.
Approaching Points
In multivariable calculus, evaluating limits requires us to consider how points in two-dimensional space approach a specific target dot. This process is known as "approaching a point," and it requires careful attention to both coordinates involved.
In this concept, it is crucial to understand both coordinate axes influence the direction to the point, meaning you might come from any direction in the plane—horizontally, vertically, or diagonally. This feature distinguishes multivariable calculus from single-variable calculus, where limits are approached along a number line.
When dealing with approaching points, it’s important to check whether the limit is consistent regardless of the path taken to get to the point. Sometimes paths can yield different limit values, which indicates the limit does not exist. Thus, for a limit to exist at a point in a two-variable function, it must result in the same value no matter the path taken as it approaches the point. However, in our case, since the function is only concerned with "x," its limit remains straightforward and path-independent.
In this concept, it is crucial to understand both coordinate axes influence the direction to the point, meaning you might come from any direction in the plane—horizontally, vertically, or diagonally. This feature distinguishes multivariable calculus from single-variable calculus, where limits are approached along a number line.
When dealing with approaching points, it’s important to check whether the limit is consistent regardless of the path taken to get to the point. Sometimes paths can yield different limit values, which indicates the limit does not exist. Thus, for a limit to exist at a point in a two-variable function, it must result in the same value no matter the path taken as it approaches the point. However, in our case, since the function is only concerned with "x," its limit remains straightforward and path-independent.
Other exercises in this chapter
Problem 56
The strength \(E\) of an electric field at point \((x, y, z)\) resulting from an infinitely long charged wire lying along the \(y\) -axis is given by \(E(x, y,
View solution Problem 57
A thin plate made of iron is located in the \(x y\) -plane. The temperature \(T\) in degrees Celsius at a point \(P(x, y)\) is inversely proportional to the squ
View solution Problem 61
For the following exercises, find the limit of the function. $$\lim _{(x, y) \rightarrow(1,2)} \frac{5 x^{2} y}{x^{2}+y^{2}}$$
View solution Problem 62
Show that the limit \(\lim _{(x, y) \rightarrow(0,0)} \frac{5 x^{2} y}{x^{2}+y^{2}}\) exists and is the same along the paths: \(y\) -axis and \(x\) -axis, and a
View solution