Problem 16 - Calculus for Biology and Medicine

Question

Show that $$\lim _{(x, y) \rightarrow(0,0)} \frac{(x-y)(x+2 y)}{x^{2}+y^{2}}$$ does not exist by computing the limit along the positive \(x\) -axis and the positive \(y\) -axis.

Step-by-Step Solution

Answer

The limit does not exist because the limits along the x-axis and y-axis are different: 1 and -2, respectively.

Step 1: Evaluate along the x-axis

When evaluating the limit along the x-axis, we set \(y = 0\). Then the limit becomes \[\lim_{x \to 0} \frac{x(x + 2 \cdot 0)}{x^2 + 0^2} = \lim_{x \to 0} \frac{x^2}{x^2} = \lim_{x \to 0} 1 = 1.\] Thus, along the positive \(x\)-axis, the limit approaches \(1\).

Step 2: Evaluate along the y-axis

When evaluating the limit along the y-axis, set \(x = 0\). The limit becomes \[\lim_{y \to 0} \frac{(0 - y)(0 + 2y)}{0^2 + y^2} = \lim_{y \to 0} \frac{-2y^2}{y^2} = \lim_{y \to 0} -2 = -2.\] Thus, along the positive \(y\)-axis, the limit approaches \(-2\).

Step 3: Conclusion on limit existence

Since the limit along the \(x\)-axis is \(1\) and the limit along the \(y\)-axis is \(-2\), and these two limits do not agree, we can conclude that the overall limit \(\lim _{(x, y) \rightarrow(0,0)} \frac{(x-y)(x+2 y)}{x^{2}+y^{2}}\) does not exist.

Key Concepts

Limits in multiple dimensions

In multivariable calculus, we often deal with functions that depend on two or more variables. These functions allow for more complex behaviors than single-variable functions. One of the critical concepts here is the limit of a function as it approaches a point in multiple dimensions. Consider a two-variable function \( f(x,y) \). The limit of \( f(x,y) \) as \((x, y)\) approaches \((a, b)\) is the value that \( f(x,y) \) gets closer to as the point \((x, y)\) gets closer to \((a, b)\).
This can be trickier than single-variable limits. There are infinitely many paths along which \((x, y)\) can approach \((a, b)\). Consequently, it’s essential to check if the function's limit remains consistent along these potentially varied paths.
The essence of a multidimensional limit is that the limit exists only if the function approaches the same value no matter the path taken towards the point.

Path-dependent limits

When evaluating limits in multiple dimensions, one key concern is whether the limit is path-dependent. This means checking if the limit is the same regardless of the path taken to approach a point. When we say the limit is path-dependent, it implies that different paths provide different limit values.
Here's where path-dependent limits can cause a function not to have a limit at a particular point. For instance, you might find that approaching along the x-axis yields a different limit than approaching along the y-axis.

• A path-dependent limit means the outcome changes based on the direction.
• If evaluated from different directions, we might witness consistency or discrepancies.
• If any two paths yield different results, then the overall limit does not exist.

Understanding path-dependence helps clarify why a function might not have a limit at a certain point, as demonstrated through our example equation evaluated along different axes.

Coordinate axis evaluation

Evaluating limits along coordinate axes simplifies understanding multivariable limits. By doing this, we essentially "slice" the multidimensional problem into one-dimensional problems, where our understanding of single-variable calculus can apply.
The approach involves setting one variable to zero and simplifying the function to observe limit behavior along the other axis. This step helps identify variations in limit values when approaching along different directions.

• Set \(y = 0\) to evaluate along the x-axis, and simplify the function.
• Set \(x = 0\) for evaluation along the y-axis.
• Compare the results from these different evaluations to check for consistency.

In the exercise, evaluating along the positive x-axis and y-axis resulted in different limits \( (1\ and\ -2) \), which confirmed the non-existence of the limit as \((x, y)\) approaches \((0,0)\). This highlights the importance of checking various paths to verify the existence or non-existence of limits in multivariable calculus.