Problem 77 - Calculus Volume 3

Question

For the following exercises, evaluate the limits at the indicated values of \(x\) and \(y\). If the limit does not exist, state this and explain why the limit does not exist. $$\lim _{(x, y) \rightarrow(0,0)} \ln \left(x^{2}+y^{2}\right)$$

Step-by-Step Solution

Answer

The limit does not exist because it approaches negative infinity at (0,0).

Step 1: Understanding the Function

Consider the function \( f(x, y) = \ln(x^2 + y^2) \). We need to evaluate its limit as \((x, y)\) approaches \((0, 0)\).

Step 2: Analyzing the Limit

We check if the limit exists by considering approaches towards \((0, 0)\) along different paths, since \((x^2 + y^2)\) describes a paraboloid.

Step 3: Attempting Direct Approach

If we substitute \((x, y) = (0, 0)\), we have \( \ln(0^2 + 0^2) = \ln(0) \), which is undefined since the logarithm of zero is negative infinity.

Step 4: Path 1: Along the x-axis

Set \( y = 0 \) and consider the path along the x-axis: \( \lim_{x \to 0} \ln(x^2) = \ln(0) = -\infty \), which is undefined.

Step 5: Path 2: Along the y-axis

Set \( x = 0 \) and consider the path along the y-axis: \( \lim_{y \to 0} \ln(y^2) = \ln(0) = -\infty \), which is again undefined.

Step 6: Conclusion

Since the limit approaches negative infinity as we come closer to \((0, 0)\) along both axes, and \( \ln(x^2 + y^2) \to \ln(0) \) which is undefined, the limit does not exist.

Key Concepts

Logarithmic Functions

Logarithmic functions are a fundamental tool in mathematics, used to solve equations involving exponential growth or decay. The general form is \( \ln(x) \), where \( \ln \) represents the natural logarithm, and it is defined only for positive numbers. This means the expression inside the logarithm must be greater than zero.

• If \( x = 1 \), then \( \ln(1) = 0 \).
• If \( x < 1 \), the logarithm yields a negative value.
• As \( x \to 0^+ \), \( \ln(x) \to -\infty \).

Understanding this behavior is crucial when evaluating limits involving logarithmic expressions, particularly near zero, where it becomes undefined due to the approach towards negative infinity.

Paraboloid

A paraboloid is a quadric surface that has sections which are parabolas. When dealing with multivariable functions, recognizing shapes like paraboloids can be very telling about the behavior of the function.
In the expression \( x^2 + y^2 \), the surface described is a circular paraboloid, which opens upwards. Its vertex is at the origin \((0, 0)\), where both \( x \) and \( y \) are zero.

• As you move away from the vertex, the value of \( x^2 + y^2 \) increases.
• Along any path approaching the origin, \( x^2 + y^2 \) trends towards zero.

For the problem at hand, it's important to note that the paraboloid reaches its minimum value at the origin \((0, 0)\). This fact, combined with the properties of logarithmic functions, is key to determining whether a multivariable limit involving this paraboloid expression exists.

Calculus Problem Solving

Multivariable calculus involves evaluating limits of functions as they approach a point in a plane or space. This is typically done by approaching the limit point along various paths.
To determine if a limit exists:

• Try substituting directly into the function, if possible.
• If direct substitution results in an undefined form, evaluate the limit along different paths.
• Verify if the result is the same along all paths to confirm the limit's existence. If differing results are obtained, the limit does not exist.

In the given problem, multiple paths (x-axis, y-axis) leading to negative infinity confirmed the non-existence of the limit, since a consistent finite limit could not be established.

Undefined Limits

Undefined limits occur when a function does not settle at a particular value as it approaches a certain point. In our exercise, the limit was undefined due to the nature of the logarithm.

• The logarithmic expression \( \ln(x^2 + y^2) \) becomes problematic as \( (x, y)\to (0, 0) \).
• Since \( x^2 + y^2 \) trends towards zero at the origin, \( \ln(0) \) creates an undefined limit due to its tendency toward negative infinity.

Undefined limits are a critical concept in exploring function behavior in calculus, highlighting paths or directions where the function ceases to converge.