Question

Use the formal definition of a limit to prove that \(\lim _{(x, y) \rightarrow(a, b)} c f(x, y)=c \lim _{(x, y) \rightarrow(a, b)} f(x, y).\)

Short answer

Question: Prove that the limit of a constant times a function equals the constant times the limit of the function as (x, y) approaches (a, b) using the ε-δ definition of a limit. Solution: Given the statement, we are asked to prove: \(\lim _{(x, y) \rightarrow(a, b)} c f(x, y) = c \lim _{(x, y) \rightarrow(a, b)} f(x, y)\) Using the steps from the provided solution, we can state the formal definition of a limit, work with absolute values and inequalities, apply the limit definition to the function \(f(x,y)\) and finally connect back to the limit of the function \(cf(x,y)\). The result is that for any \(\epsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < \sqrt{(x-a)^{2}+(y-b)^{2}} < \delta\), then \(|cf(x, y) - cL| < \epsilon\). This proves the desired statement using the ε-δ definition of a limit.

Step-by-step solution

State the formal definition of a limit

In order for us to prove the given statement, we need to recall the formal definition of a limit. The limit of a function \(f(x,y)\) as \((x, y)\) approaches \((a, b)\) is \(L\), denoted as \(\lim _{(x, y) \rightarrow(a, b)} f(x, y) = L\), if for any \(\epsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < \sqrt{(x-a)^{2}+(y-b)^{2}} < \delta\), then \(|f(x, y) - L| < \epsilon\).

Define the given equation in terms of the limit definition

We need to prove that \(\lim _{(x, y) \rightarrow(a, b)} c f(x, y) = c \lim _{(x, y) \rightarrow(a, b)} f(x, y)\), where \(c\) is a constant. Let's denote the limit of \(f(x, y)\) as \((x, y) \rightarrow(a, b)\) as \(L\). We want to show that the limit of \(cf(x,y)\) as \((x, y) \rightarrow(a, b)\) is equal to \(cL\). Using the limit definition, we need to show that for any \(\epsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < \sqrt{(x-a)^{2}+(y-b)^{2}} < \delta\), then \(|cf(x, y) - cL| < \epsilon\).

Work with absolute values and inequalities

Our goal is to show that \(|cf(x, y) - cL| < \epsilon\). By factoring out the constant \(c\) from the expression, we can rewrite it as \(|c||f(x, y) - L|\). Since \(c\) is a constant, we can divide \(\epsilon\) by \(|c|\) to obtain a new value \(\epsilon' = \frac{\epsilon}{|c|}>0\).

Apply the limit definition to \(f(x, y)\)

Since \(\lim _{(x, y) \rightarrow(a, b)} f(x, y) = L\), given any \(\epsilon' > 0\), there exists a \(\delta > 0\) such that if \(0 < \sqrt{(x-a)^{2}+(y-b)^{2}} < \delta\), then \(|f(x, y) - L| < \epsilon'\).

Connect back to the limit of \(cf(x,y)\)

We now have that \(|f(x, y) - L| < \epsilon'\). Now use the fact that \(\epsilon' = \frac{\epsilon}{|c|}\): \(|f(x, y) - L| < \frac{\epsilon}{|c|} \implies |c||f(x, y) - L| < \epsilon\). Since we defined \(|c||f(x, y) - L|\) as \(|cf(x, y) - cL|\), we can say that for any \(\epsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < \sqrt{(x-a)^{2}+(y-b)^{2}} < \delta\), then \(|cf(x, y) - cL| < \epsilon\). Therefore, according to the formal definition of a limit, we have proven that \(\lim _{(x, y) \rightarrow(a, b)} c f(x, y) = c \lim _{(x, y) \rightarrow(a, b)} f(x, y)\).

Key concepts

Formal Definition of Limit

The formal definition of a limit is a fundamental concept in calculus, particularly useful for understanding how functions behave as they approach certain points. It formalizes the intuitive notion that values of a function can get arbitrarily close to a given number as the input approaches a certain point. In the context of multivariable calculus, this is expanded from single-variable functions to functions of two or more variables, such as \(f(x, y)\).

The formal definition says: the limit of a function \(f(x, y)\) as \((x, y)\) approaches \((a, b)\) is \(L\), if for every \(\epsilon > 0\) there exists a \(\delta > 0\) such that whenever \(0 < \sqrt{(x-a)^2+(y-b)^2} < \delta\), it follows that \(|f(x, y) - L| < \epsilon\).

This definition extends the single-variable concept to situations where direction and distance between points matter, reflecting the behavior of functions in more than one dimension. Understanding this is crucial in building the groundwork for rigorous proofs and applications in multivariable calculus.

Multivariable Calculus

Multivariable calculus expands the ideas of calculus to functions with multiple inputs, typically involving two or three dimensions. This branch of calculus is vital for fields like physics, engineering, and economics, where situations often depend on more than just a single variable.

With multivariable calculus, we analyze how a function behaves when input variables change simultaneously. For example, in a function \(f(x, y)\), changes in both \(x\) and \(y\) are studied to ascertain the function's behavior nearing a particular point \((a, b)\). This requires an understanding of not just limits and derivatives, but also gradients and partial derivatives, which describe how a function changes with respect to each variable individually.

It introduces new concepts such as level curves for functions of two variables, which help to visualize the function's behavior. By exploring these multi-dimensional applications, multivariable calculus allows for a deeper examination of more complex systems and phenomena.

Continuity and Convergence

Continuity and convergence are interconnected concepts frequently discussed in the context of limits and multivariable calculus.

A function is called continuous at a point \((a, b)\) when the value of the function at \((a, b)\) matches the value of the limit as it approaches \((a, b)\). In other words, \(f(x, y)\) is continuous at \((a, b)\) if \(\lim_{(x, y) \to (a, b)} f(x, y) = f(a, b)\). This ensures that no abrupt changes occur in the graph of the function as we move through that point, which is crucial for understanding and predicting the function's behavior.

Convergence involves the idea that a sequence of numbers generated by a function will approach a particular value as the input gets closer to a specific point. In multivariable calculus, this extends to sequences of points in a space converging towards a limit point. For example, as \((x, y)\) gets arbitrarily close to \((a, b)\), the sequence of \(f(x, y)\) values converges to \(L\), illustrating the predictable nature of the function near that point.

Both continuity and convergence assure that functions behave in a stable and predictable manner, which is fundamental for advanced calculus topics and their practical applications.