Problem 37 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(f(x, y)=x^{2} y /\left(x^{4}+y^{2}\right)\). (a) Show that \(f(x, y) \rightarrow 0\) as \((x, y) \rightarrow(0,0)\) along any straight line \(y=m x\). (b) Show that \(f(x, y) \rightarrow \frac{1}{2}\) as \((x, y) \rightarrow(0,0)\) along the parabola \(y=x^{2}\) (c) What conclusion do you draw?

Step-by-Step Solution

Answer

The limit as \((x, y) \to (0,0)\) doesn't exist; it's path-dependent.

Step 1: Substitute Line Equation in Part (a)

Consider the straight line given by the equation \(y = mx\). Substitute \(y = mx\) into the function to get: \[ f(x, mx) = \frac{x^2 (mx)}{x^4 + (mx)^2} = \frac{mx^3}{x^4 + m^2x^2} = \frac{mx^3}{x^2(x^2 + m^2)}. \] Simplify this to \( \frac{mx}{x^2 + m^2} \).

Step 2: Evaluate Limit in Part (a)

Now, evaluate the limit as \((x, mx) \to (0,0)\):\[ \lim_{x \to 0} \frac{mx}{x^2 + m^2} = \frac{m \cdot 0}{0 + m^2} = 0. \] So, \( f(x, y) \to 0 \) as \((x, y) \to (0, 0)\) along any line \(y = mx\).

Step 3: Substitute Parabola Equation in Part (b)

For the parabola \(y = x^2\), substitute into the function \[ f(x, x^2) = \frac{x^2 (x^2)}{x^4 + (x^2)^2} = \frac{x^4}{x^4 + x^4} = \frac{x^4}{2x^4} = \frac{1}{2} \].

Step 4: Evaluate Limit in Part (b)

Evaluate the limit as \((x, x^2) \to (0,0)\):\[ \lim_{x \to 0} \frac{1}{2} = \frac{1}{2}. \] Thus, \( f(x, y) \to \frac{1}{2} \) as \((x, y) \to (0,0)\) along the parabola \(y = x^2\).

Step 5: Conclusion from Part (c)

The limit of \( f(x, y) \) as \((x, y) \to (0, 0)\) depends on the path taken to approach the origin. Different paths give different limits: zero along any straight line \( y = mx \) and \( \frac{1}{2} \) along the parabola \( y = x^2 \). This indicates that the limit of \( f(x, y) \) does not exist at \((0,0)\) as it is path-dependent.

Key Concepts

Path-dependent Limits

In multivariable calculus, understanding path-dependent limits is crucial. Simply put, path-dependent limits occur when the value of a limit changes depending on the path taken to approach a point. In contrast to single-variable calculus, where approaching a point typically means moving along a single line, multivariable calculus allows for infinite paths. Each of these paths can yield a different result.
In the problem example given, the function \(f(x, y) = \frac{x^2 y}{x^4 + y^2}\) shows different outcomes based on diverse paths. As we approach the origin \((0,0)\) along any straight line \(y = mx\), the limit is zero. However, if we approach along a parabolic path \(y = x^2\), the limit becomes \(\frac{1}{2}\).
This variation is a hallmark of path-dependent limits. It tells us that the overall limit at a point, in this case \((0,0)\), does not exist because we do not get a consistent value from all possible paths. Recognizing such situations helps in analyzing the behavior of complex functions around critical points.

Evaluating Limits Along Different Paths

To evaluate limits along various paths in multivariable calculus, you often substitute a path equation into the given function. This substitution helps to simplify the expression so you can find the limit more straightforwardly.

In our step-by-step example, the two paths analyzed were:

• Straight lines described by \(y = mx\).
• A parabola described by \(y = x^2\).

For straight lines, substituting \(y = mx\) into the function, we simplify to \(\frac{mx}{x^2 + m^2}\). When evaluating the limit as \((x, y) \to (0, 0)\), it consistently results in zero.
On the other hand, for the parabola \(y = x^2\), the substitution simplifies the function to a constant \(\frac{1}{2}\). Hence, as \((x, y) \to (0, 0)\), the limit is \(\frac{1}{2}\).
Each path's unique approach shows how substituting different path equations reveals varied limit behavior at the same point. It's a practical and systematic approach to evaluating limits where multiple variables are involved.

Limit of Multivariable Functions

The limit of a multivariable function aims to determine if a particular value exists as we approach a point from various paths. If all paths lead to the same limit, the limit exists. If not, it is path-dependent and does not exist in a definitive sense.

In our example function \(f(x, y) = \frac{x^2 y}{x^4 + y^2}\), when approaching the origin \((0,0)\), the different values along specific paths indicate that the overall limit is path-dependent.
Key points related to limits of multivariable functions include:

• To check if a limit exists, explore multiple paths to see if they converge to the same value.
• If any two paths give different results, the limit does not exist at that point.

The discrepancy in outcomes across paths implies that a unique limit cannot be established at the origin. This highlights the complexity and fascinating nature of multivariable calculus, where understanding limits extends beyond a single axis into an entire plane.