Problem 77 - Calculus 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 and diverges to \(-\infty\).

Step 1: Understanding the Function

The function is given as \( \ln(x^2 + y^2) \). The expression inside the logarithm, \( x^2 + y^2 \), represents a sum of squares of \( x \) and \( y \). As both \( x \) and \( y \) approach zero, the expression \( x^2 + y^2 \) approaches zero.

Step 2: Properties of Logarithm

The natural logarithm \( \ln(z) \) is defined only for positive real numbers. As \( z \) approaches zero from the positive side, \( \ln(z) \) approaches negative infinity. In this case, \( x^2 + y^2 \) approaches zero, meaning that the argument of the logarithm is positive and approaches zero.

Step 3: Limit Evaluation

As \( (x, y) \rightarrow (0,0) \), the expression within the logarithm, \( x^2 + y^2 \), approaches zero. Therefore, \( \ln(x^2 + y^2) \rightarrow -\infty \). The closer the point \((x, y)\) gets to \( (0, 0) \), the smaller \( x^2 + y^2 \) gets, which makes the logarithm tend towards negative infinity.

Step 4: Conclusion

Since \( \ln(x^2 + y^2) \) tends to \(-\infty\), the limit \( \lim_{(x, y) \to (0, 0)} \ln(x^2 + y^2) \) does not exist in the sense of approaching a real number. Instead, it diverges to \(-\infty\). Therefore, the limit does not exist.

Key Concepts

Natural Logarithm Properties

The natural logarithm, denoted as \( \ln(z) \), is a fundamental mathematical function that plays a crucial role in calculus and, particularly, in examining limits. A few key properties include:

• **Domain**: The natural logarithm is only defined for positive real numbers. This means \( z > 0 \).
• **Behavior near Zero**: As a positive number \( z \rightarrow 0^+ \), \( \ln(z) \rightarrow -\infty \). This important trait causes the logarithmic function to dip sharply towards negative infinity as the input diminishes.
• **Monotonicity**: The natural logarithm is a strictly increasing function. As the input \( z \) increases, so does \( \ln(z) \).

These properties are essential when evaluating limits, such as those involving functions like \( \ln(x^2 + y^2) \). Here, since \( x^2 + y^2 > 0 \) as \( x \) and \( y \) approach zero, we deal with the situation where the argument approaches a point close to zero, leading the logarithm towards \(-\infty\). It is critical to identify when such behavior causes the limit to diverge.

Limit Divergence

When dealing with multivariable limits, understanding the notion of limit divergence is crucial. A limit is said to diverge when it does not approach a single finite value. But rather, it tends towards infinity or negative infinity. In the exercise at hand, the function \( \ln(x^2 + y^2) \) demonstrates limit divergence.As \( (x, y) \rightarrow (0,0) \), the expression \( x^2 + y^2 \) itself tends to zero from the positive direction. This result implies that the argument for the logarithm becomes infinitesimally small but positive. Consequently, given the properties of logarithms, \( \ln(x^2 + y^2) \) heads towards \(-\infty\).

• **Infinity vs Real Numbers**: In calculus, a finite limit requires the value to approach a real number. If it reaches any form of infinity, it is typically classified as divergent.
• **Behavior Identification**: Correctly recognizing limits that grow indefinitely negative or positive is key to solving such limit problems correctly.

Thus, understanding this divergence helps clarify why the limit in the exercise does not exist in the traditional sense.

Function Behavior Near Origin

The behavior of multivariable functions near the origin (or any point) is a fundamental concept in calculus, especially when evaluating limits. This involves analyzing how the function behaves as its variables approach specific values.For the function \( \ln(x^2 + y^2) \) given in the problem, its behavior is dictated by how the expression \( x^2 + y^2 \) approaches zero as \( x \) and \( y \) inch closer to the origin:

• **Approaching Zero**: The squared terms \( x^2 \) and \( y^2 \) ensure non-negativity, meaning that \( x^2 + y^2 \geq 0 \), which leads the sum to always be positive as long as \( (x, y) eq (0,0) \).
• **Logarithm Behavior**: Near the origin, \( x^2 + y^2 \rightarrow 0^+ \) causes \( \ln(x^2 + y^2) \) to plunge towards \(-\infty\).

Understanding this behavior near the origin is crucial for accurately determining how the function behaves, particularly in identifying cases of limit divergence.