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Understanding limits in multivariable calculus can be a bit tricky compared to single-variable limits. Here, we deal with functions of more than one variable, such as functions that depend on both \(x\) and \(y\). When finding limits of multivariable functions, you want to determine how the function behaves as it approaches a specific point. In our exercise, that point is \((0,0)\).

Unlike single-variable calculus, multivariable functions approach the limit from infinitely many directions. Instead of just approaching from the left or right as in one-dimensional cases, in two dimensions, you can approach from any angle or path.

• The strategy is often to substitute different paths (lines, curves, etc.) that intersect the point of interest.
• If the limits along all possible paths yield the same result, the limit exists at that point. If different paths provide different limits, the limit is said to not exist.

In this exercise, we substitute paths along lines of form \(y=mx\) and a parabola \(x=y^2\) and see how they affect the limit.

Path dependence is a critical concept when examining limits of multivariable functions. It investigates how the choice of path affects the outcome of the limit.

In single-variable calculus, you only approach a point from two directions. However, in multivariable calculus, path dependence highlights how approaching a point along different paths could lead to different limits.

• This occurs because the function might behave differently along various paths due to its interaction with both \(x\) and \(y\).
• The existence of a multivariable limit requires that the limit be the same along every possible path approaching the point.
• To check for path dependence, you can substitute different paths into the function and compare the resulting limits.

In our example, different paths gave different limits: one was 0 when using lines \(y=mx\), and \(\frac{3}{2}\) using the parabola \(x=y^2\), demonstrating the path-dependent nature of the function.

To determine if the limit exists for our function as it approaches \((0,0)\), we use specific paths by substituting expressions into the function. Path substitution simplifies the original function by reducing it to a single-variable limit.

For lines, we substitute \(y=mx\):

• The function becomes \(\frac{3m^2x^4}{x^3+m^6x^6}\), which simplifies to \(\frac{3m^2x}{1+m^6x^3}\). As \(x\) approaches 0, this expression approaches 0.
• This means the function's limit along any line \(y=mx\) is 0.

For the parabolic path given by \(x=y^2\):

• The substitution leads to \(\frac{3(y^2)^2y^2}{(y^2)^3 + y^6}\), simplifying straight to \(\frac{3}{2}\).
• This indicates a different limit, \(\frac{3}{2}\), along the parabola \(x=y^2\).

With different limits from the linear and parabolic paths, we conclude that the overall limit does not exist at the point \((0,0)\), emphasizing the importance of checking different paths in multivariable calculus.