Whether or not things are "undefined" largely depeneds on what framework you are working in.
If we are working in the naturals, we might say that $3-5$ is undefined.
There are many systems where it makes sense to assign $\frac{n}{0}$ some value. In this particular example, it is defined to be complex infinity, which can be thought of as follows: suppose we are looking at the complex plane. Similarly to how the complex number "0" is represented by a zero vector of arbitrary direction, we wish to associate all complex numbers of infinite absolute value (regardless of direction) to a single point.
This is complex infinity, and geometrically, by associating all complex numbers of infinite absolute value to be the same on the plane, we have formed a sphere, one with zero on the bottom, and complex infinity on the top. This is called the Riemann Sphere.
WolframAlpha understands the expression $\sqrt[3]{x}$ for negative x in a different way than you expect.
Try this: lim\frac{\sqrt{1-x}-3}{2+surd(x,3)} as x to -8
Best Answer
WA's
ComplexInfinityis the same as Mathematica's: it represents a complex "number" which has infinite magnitude but unknown or nonexistent phase. One can useDirectedInfinityto specify the phase of an infinite quantity, if it approaches infinity in a certain direction. The standardInfinityis the special case of phase 0. Note thatInfinityis different fromIndeterminate(which would be the output of e.g.0/0).Some elucidating examples:
0/0returnsIndeterminate, since (for instance) the limit may be approached as $\frac{1/n}{1/n}$ or $\frac{2/n}{1/n}$, resulting in two different real numbers.1/0returnsComplexInfinity, since (for instance) the limit may be approached as $\frac{1}{-1/n}$ or as $\frac{1}{1/n}$, but every possible way of approaching the limit gives an infinite answer.Abs[1/0]returnsInfinity, since the limit is guaranteed to be infinite and approached along the real line in the positive direction.In your particular example, you get
ComplexInfinitybecause the infinite limit may be approached as (e.g.) $n^n$ or as $n^{n+i}$.