Question

Explain what $\underset{(x,y)\to (a,b)}{lim}f(x,y)=L$ means.

Short answer

Question: Explain the meaning of the limit of a function in two variables, f(x, y), as the point (x, y) approaches another point (a, b) with the given limit L. Answer: The limit of a function f(x, y) as (x, y) approaches (a, b) means that the function approaches a value L as we get close to the point (a, b) in the xy-plane. No matter how close we want the function values to be to the limit L, we can always find a small enough region around the point (a, b) such that whenever (x, y) is in that region (but not equal to (a, b)), the function value will be within the desired closeness to L. This implies that as we get closer and closer to the point (a, b), the function f(x, y) gets arbitrarily close to L.

Step-by-step solution

Understanding the limit of a function in two variables

A limit of a function f(x, y) as (x, y) approaches (a, b) tells us about the behavior of the function near the point (a, b). Specifically, it tells us that the function approaches a value L as we get close to the point (a, b) in the xy-plane.

Defining the limit in terms of the distance between the points

The limit can be defined formally by saying that for any small positive number, denoted by $ϵ$, there exists another small positive number, denoted by $\delta$, such that if the distance between the points (x, y) and (a, b) is less than $\delta$, then the absolute difference between the values of the function f(x, y) and L is less than $ϵ$. Mathematically, this can be written as: $$\underset{(x,y)\to (a,b)}{lim}f(x,y)=L\text{ if }0<\sqrt{(x-a{)}^2+(y-b{)}^2}<\delta ⟹|f(x,y)-L|<ϵ$$

Interpreting the meaning of the limit

The previous definition means that no matter how close we want the function values to be to the limit L, we can always find a small enough region around the point (a, b) such that whenever (x, y) is in that region (but not equal to (a, b)), the function value will be within the desired closeness to L. In other words, as we get closer and closer to the point (a, b), the function f(x, y) gets arbitrarily close to L.

Key concepts

Multivariable Calculus

Multivariable calculus is an extension of single-variable calculus to functions of several variables, in this case, two variables, x and y. When working with functions of two variables, the visualization and interpretation of concepts require a 3-dimensional perspective. Imagine a surface in three dimensions, where any point on the surface is determined by two coordinates, x and y, and the height at that point is given by the function's value, f(x, y).

The study of limits in this context helps us understand how the function behaves near a point, even if the function isn't explicitly defined at that point. For example, understanding how a hill's slope changes as you approach its peak or how temperature varies as you move across a room. The cornerstone of multivariable calculus involves analysis of curves, surfaces, and their motion through space, all of which are essential in physics, engineering, economics, and beyond.

Epsilon-Delta Definition

The epsilon-delta definition is the formal, mathematical way of defining what we mean by 'limit of a function.' Specifically, when studying limits in the realm of two variables, this definition helps in quantifying the idea of 'closeness.'

Let's break it down: \

\
• $ϵ$ (epsilon) represents how close the function values need to be to the limit L. Think of \ as your level of tolerance for deviation from the expected outcome, L.
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• $\delta$ (delta) represents a distance radius around the point (a, b) such that if (x, y) is inside this delta-radius but not exactly at (a, b), then f(x, y) is within our epsilon tolerance of L.
\

\This rigorous definition allows us to discuss the precise behavior of functions and is indispensable when proving theorems in calculus or analyzing data with minute precision in fields like engineering and physical sciences.

Continuity of Functions

Continuity is among the most vital concepts in calculus, as it describes functions that are unbroken or smooth. For a function of two variables, we say the function is continuous at a point (a, b) if, and only if, the limit of the function as (x, y) approaches (a, b) equals the function's value at that point. In simpler terms, you can draw the function at that point without lifting your pen off the graph paper.

Continuity has practical implications. Consider, for example, designing a roller coaster—continuity ensures that the path is smooth, so riders don't experience sudden, jolting changes in direction. In multivariable functions, the idea extends to a surface where there are no holes, jumps, or vertical walls when you move infinitesimally close to any point on the surface. Ensuring continuity in functions is essential when modeling real-world phenomena like airflow over a wing or the flow of a fluid, where sudden changes can have drastic effects.