91Ó°ÊÓ

When we talk about a limit not existing, it means that no matter how hard we try, we cannot find a single value that a function approaches as we move closer to a certain point. In the world of multivariable calculus, things are a bit more complex because we have extra dimensions to think about. Imagine you are in a busy city with roads leading in every direction. It's hard to agree on a central spot if everyone is coming from different streets and getting surprised by all sorts of distractions.

In the problem, as we approached the point \((2, 1, 0)\), different paths towards this point gave us different answers. This inconsistency in limits along paths is a common telltale sign of non-existence. The key element here was the denominator becoming zero, creating a situation where normal numerical values turn into very problematic results, such as dividing by zero. This is what leads to the conclusion that the limit simply does not exist.

Three-dimensional limits add an extra layer of complexity since we are dealing with not just one or two, but three variables at once. This means we need to consider how the function behaves as it approaches a particular point in a three-dimensional space. Imagine a sculptor carving a statue. Each angle and approach gives a different perspective of the sculpture—even though it is the same work of art.

In our problem, we had the function \(\frac{(x+y+z-3)^{5}}{z^{3}(x-2)(y-1)}\) with \((x, y, z)\) approaching \((2, 1, 0)\). As you can imagine, checking from this three-dimensional perspective gave us insight into why the limit might not exist. The denominator \(z^3(x-2)(y-1)\) simultaneously short-circuits the process when any of those terms become zero.

• If \(x = 2\), \(y = 1\), or \(z = 0\), the whole denominator becomes zero.
• This leads to an undefined form, much like a statue with pieces missing or out of balance.

"Path dependence" refers to how the path you take in approaching a point can affect the outcome—especially in multivariable limits. If different paths result in different limits, then no single limit value can be confirmed at that point. Think of it like trying to meet a friend in a big park; if you both approach the meeting point from different sides but get different views and obstacles, it can be confusing whether you’re at the same spot.

In this exercise, we found different limits based on the paths we took towards \((2, 1, 0)\). For example:

• One path yielded a limit of \(0/0\), which is indeterminate.
• Another path clearly gave us \(1\), a determined end value.

These varying results indicate that there is no single, consistent limit for this function at the given point. The conflict in results across different paths reinforces the understanding that the limit doesn't exist because the approach can change the outcome dramatically.