Question

Consider \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2} y}{x^{4}+y^{2}}\). (a) Determine (if possible) the limit along any line of the form \(y=a x\) (b) Determine (if possible) the limit along the parabola \(y=x^{2}\). (c) Does the limit exist? Explain.

Short answer

The limit does not exist because it is not the same along all paths approaching (0,0). More specifically, the limit is 0 along any line \(y = ax\), but it is 1/2 along the parabola \(y = x^2\).

Step-by-step solution

Determine the limit along the line y=ax

Substitute \(y = ax\) into the original limit expression. This results in: \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2} ax}{x^{4}+a^{2}x^{2}}\)\n Next, simplify the expression by dividing both the numerator and the denominator by \(x^2\) (notice that this is possible because \(x^2\) is nonzero). This gives: \(\lim _{x \rightarrow 0} \frac{a x}{x^{2}+a^{2}} = 0\). This is the limit along the line \(y = ax\).

Determine the limit along the parabola \(y = x^2\)

Substitute \(y = x^2\) into the original limit expression. This results in: \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2} x^{2}}{x^{4}+x^{4}}\)\n Then simplify this to \(\lim _{x \rightarrow 0} \frac{1}{2} = \frac{1}{2}\). This is the limit along the parabola \(y=x^2\).

Determine if the limit exists

For the limit to exist, it needs to be the same regardless of the path taken to (0,0). However, as we saw in the first two steps, the limit is 0 along any line \(y = ax\), but it is 1/2 along the parabola \(y = x^2\). Because these two are not the same, the limit does not exist.

Key concepts

Limits of Functions

In multivariable calculus, the concept of limits of functions is an extension of limits from single-variable calculus. It concerns the behavior of a function as its input variables approach some value. The limit of a function at a point essentially captures what value the function is expected to approach as the input gets infinitely close to that point.

The formal definition of a limit for a multivariable function is more complex than for a single-variable function because you can approach the point from infinitely many paths in the domain. When finding limits, it's important to remember that if a limit exists, the function will approach the same value no matter from which path the point is approached.

Limit Along a Line

A limit along a line is a specific case of evaluating the limit of a multivariable function. Instead of considering all possible paths to approach a point, one only looks at linear paths that approach the point. In the context of the given problem, we considered the line with the equation y = ax which gives us a family of lines passing through the origin.

To compute the limit along a line, we substitute the linear equation into the function's formula and then compute the limit as one variable approaches zero. This method, as shown in the original solution, simplifies the expression which can then be evaluated in a way that's similar to single-variable limits.

Limit Along a Parabola

When evaluating a limit along a parabola, we follow a similar process as the limit along a line but using a parabolic path instead. For the given problem, the path chosen is the parabola described by y = x^2. By substituting y with x^2 in the function, we transform the original multivariable function into a single-variable function where we can then take the limit as x approaches zero.

While a parabola is just one type of curve amongst many potential paths in multivariable limits, it illustrates that different paths can affect the outcome of a limit, hinting at the existence of path dependence in these kinds of limits.

Path Dependence

Path dependence in multivariable calculus is an essential concept highlighting the fact that the value of the limit of a function can depend on the path taken to approach the point under consideration. The example from this exercise beautifully illustrates that when approaching the origin along different paths (a line and a parabola), we get different limit values.

Path dependence is crucial when determining the existence of a limit. If we find at least two different paths that yield different limit values, we can say that the limit does not exist because the value it approaches should be independent of the path taken.

Existence of Limits

The existence of limits is governed by whether a function approaches a specific value regardless of the path taken as it gets close to a point. A limit exists only if every path towards the point results in the same limit value. In the given exercise, we discovered that the limit approaches different values depending on whether we approached the origin along a line or a parabola. This inconsistency means that the limit of the function as (x, y) → (0, 0) does not exist.

The failure to meet this condition is critical when assessing multivariable functions. It guides us to understand the behavior of functions in the vicinity of a point and is foundational to further studies in continuity and differentiability in multivariable calculus.