Problem 19 - Calculus for Biology and Medicine

Question

Compute $$ \lim _{(x, y) \rightarrow(0,0)} \frac{2 x y}{x^{3}+y x} $$ along lines of the form \(y=m x\), for \(m \neq 0\), and along the parabola \(y=x^{2} .\) What can you conclude?

Step-by-Step Solution

Answer

The limit does not exist as it depends on the path taken.

Step 1: Parametrize Along Line

To compute the limit along lines of the form \(y = mx\), substitute \(y = mx\) into the expression: \[ \frac{2xy}{x^3 + yx} = \frac{2x(mx)}{x^3 + (mx)x} = \frac{2mx^2}{x^3 + mx^2}. \] Simplify the denominator to get \(x^2(x+m)\), leading to: \[ \frac{2mx^2}{x^2(x+m)} = \frac{2m}{x+m}. \] As \(x \rightarrow 0\), the expression becomes \(\frac{2m}{m} = 2\).

Step 2: Parametrize Along Parabola

Now consider \(y = x^2\) along the parabola. Substitute \(y = x^2\) into the expression to obtain: \[ \frac{2x(x^2)}{x^3 + (x^2)x} = \frac{2x^3}{x^3 + x^3} = \frac{2x^3}{2x^3} = 1. \] As \(x \rightarrow 0\), the limit is \(1\).

Step 3: Analyze Results

The limits obtained are dependent on the path chosen to approach \((0,0)\). Along the line \(y=mx\), the limit is 2, but along the parabola \(y=x^2\), the limit is 1. Since the limits are different for different paths, the overall limit does not exist.

Key Concepts

Parametric Equations

Parametric equations are a great way to describe a curve using parameters. Instead of using regular Cartesian coordinates, we express the variables in terms of a third variable, usually denoted as \(t\). This parameter \(t\) typically varies over some interval of numbers.
In our exercise, when approaching the origin along various paths such as a line or a parabola, we parametrize the equations. For lines of the form \(y = mx\), this means substituting \(y = mx\) directly into the function. Here, \(m\) is a constant, and \(x\) represents the parameter.

• Parametric equations simplify the complexity of solving multivariable limits by reducing it to a function of a single variable.
• They help in analyzing how the function behaves as it approaches a certain point along a specific path.

Understanding parametric equations can help solve complex multivariable problems by narrowing them down to simpler one-variable ones.

Path-Dependent Limits

In multivariable calculus, limits can be tricky because they may depend on the path taken to the point in question. If you approach a point \((a, b)\) along different paths and end up with different limits, then the overall limit does not exist.
In the given exercise, we took two paths to approach the point \((0,0)\): one via a straight line and another via a parabolic curve.

• When approaching along the line \(y=mx\), the limit was calculated to be 2.
• However, when approaching along the parabola \(y=x^2\), the limit turned out to be 1.

The difference in values for different paths indicates a path-dependent limit. If the limit were the same along all paths, we would say it was path-independent. In this case, since the values differ, the overall limit at that point is undefined.

Limits in Calculus

Limits are foundational to calculus, helping describe the behavior of functions as they approach specific points. In single-variable calculus, we usually consider just one path into a point along the number line. In multivariable calculus, however, things get more complex.
Imagine you want to find a limit at point \((x, y)\), like the exercise shows. There are infinite paths to consider. If using different paths results in different limit values, the overall limit does not exist.

• Limits help in understanding the continuity and boundedness of a function.
• They are essential in determining differentiability, as the notion of derivatives builds on them.
• Check the limits along varied paths to ensure consistency, verifying if the function is truly continuous around certain points.

Recognizing that an overall limit can be explored along multiple paths is crucial. It lets you confidently analyze the behavior and properties of functions beyond one dimension.