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How to determine if there is a limit or not in a function?

For example, when a limit approaches 0: x^2-5x+7/ x^3


Since we cannot divide by zero, we have a
vertical asymptote at x=0
As x→0 from the left you get x as a tiny negative
number.  This approaches 7/(very small negative)
so it drops towards negative infinity...
As x→0 from the right you get x as a tiny positive
number.  This approaches 7/(very small positive)
so it rises toward positive infinity.
as x→0 this function approaches 7/0
which is undefined.

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