Problem 1 - Calculus Early Transcendentals

Question

Suppose that \(\lim _{(x, y \rightarrow(3,1)} f(x, y)=6 .\) What can you say about the value of \(f(3,1) ?\) What if \(f\) is continuous?

Step-by-Step Solution

Answer

If \( f \) is continuous, \( f(3,1) = 6 \); otherwise, \( f(3,1) \) can be any value.

Step 1: Understanding Limits

We are given that \( \lim_{(x, y) \to (3,1)} f(x, y) = 6 \). This means as \( (x, y) \) approaches \( (3,1) \), the function \( f(x, y) \) approaches the value 6. However, this does not directly tell us the value of \( f(3,1) \).

Step 2: Considering Discontinuity

If the function \( f \) might be discontinuous at \( (3,1) \), there is no requirement for \( f(3,1) \) to equal the limit \( 6 \). Thus, \( f(3, 1) \) could be any value if \( f \) is not continuous.

Step 3: Analyzing Continuity

If the function \( f \) is continuous at the point \( (3,1) \), then by the definition of continuity, \( f(3,1) \) must equal \( \lim_{(x, y) \to (3,1)} f(x, y) = 6 \). Thus, if \( f \) is continuous, \( f(3,1) = 6 \).

Key Concepts

Understanding Continuity in Calculus

In calculus, the concept of continuity plays a vital role. A function is considered continuous at a point if, intuitively speaking, you can draw the function's graph at that point without lifting your pencil. More formally, a function \( f(x, y) \) is continuous at a point \((a, b)\) if

• \( \lim_{(x, y) \to (a, b)} f(x, y) \) exists,
• \( f(a, b) \) is defined, and
• \( \lim_{(x, y) \to (a, b)} f(x, y) = f(a, b) \).

This means, for a function to be continuous at a point, the value of the function at that point must equal the value the function approaches as the input nears that point. Applying this to the exercise, if the function \( f \) is continuous at \((3, 1)\), then we can confidently say \( f(3, 1) = 6 \). Continuity ensures a smooth behavior of functions at specified points, which is essential for solving problems in calculus without unexpected jumps or breaks.

Unpacking Discontinuity

Discontinuity in a function occurs when there is an interruption or a sudden change in the function's behavior at a particular point. This often results from the limit of the function not equalling the function's value at that point. When a function \( f(x, y) \) is discontinuous at a point \((a, b)\), one or more of the conditions for continuity fail.
There are different types of discontinuities:

• Jump Discontinuity: The function has a sudden "jump" at the point.
• Infinite Discontinuity: The function's value approaches infinity around the point.
• Removable Discontinuity: The limit exists and the function can be redefined at the point to be continuous.

In the given exercise, if the function \( f \) is discontinuous at \((3, 1)\), \( f(3, 1) \) does not necessarily equal the limit \( 6 \). It can take any value regardless of the value to which the function converges as \((x, y)\) approaches \((3, 1)\). This flexibility can sometimes be useful in problem-solving or modeling real-world situations where abrupt changes are expected.

Exploring Multivariable Functions

Multivariable functions, as the name implies, are functions with more than one input variable. They can be of the form \( f(x, y) \), where both \( x \) and \( y \) contribute to the output. These functions are fundamental in higher-dimensional analysis, allowing us to model complex systems such as physical phenomena and economic models.
The concept of limits for multivariable functions is a key topic in calculus. In these functions, we evaluate how the function behaves as the variables approach certain values from all possible directions. This can be a bit more involved than single-variable limits because the paths of approach are infinite.

• The limit \( \lim_{(x, y) \to (a, b)} f(x, y) \) exists if from whatever direction \((x, y)\) approaches \((a, b)\), \( f(x, y) \) approaches the same value.
• The analysis of multivariable limits is crucial for understanding the function’s continuity or discontinuity at a point.

In the context of the original exercise, since \( \lim_{(x, y) \rightarrow(3,1)} f(x, y)=6 \), it means the function \( f(x, y) \) approaches 6 from all directions. This insight helps establish the function's behavior near that point, aiding in comprehending the nature and potential discontinuities when unable to assure continuity. Multivariable calculus equips students with the tools needed to explore and analyze such complex scenarios.