Limits are...weird...for multi-variable functions | Limits along paths
In single variable calculus, you only had to take a limit from the left and from the right. In multi variable calculus, you can approach from every single direction. One way to say a limit doesn't exist is then to find two different paths where along those two different paths the limit approaches different values. Then for sure the limit of the function does not exist. However, you could find 10,100,1000 paths where the limit was the same along every path and yet the limit still didn't exist because you just hadn't found the one path that was weird yet. This is problematic and motivates upgrading the very definition of the limit from what we saw in first year calculus.
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