Changes

So infinity might arise in statements like these:

So infinity might arise in statements like these:

*<math>\frac{1}{0} = \infty\ \ </math>  Mathematics does not allow division by zero, so this is technically not allowed unless does as a limit, as follows:

*<math>\frac{1}{0} = \infty\ \ </math>  Mathematics does not allow division by zero, so this is wrong.

*<math>\lim_{x\to 0}\frac{1}{x} = \infty\ \ </math>  Yes, one could say that "the limit is infinite," since that includes both positive and negative infinite values.  But to say that the limit is "equal to infinity," one would have to say that this is the limit from the right. The limit from the left is "minus infinity," that is, unboundedly negative. This sort of statement, in terms of limits, is what was presumably meant by the incorrect statement above.

*<math>\lim_{x\to 0}\frac{1}{x} = \infty\ \ </math>  Yes, one could say that "the limit is infinite," since that includes both positive and negative infinite values.  But to say that the limit is "equal to infinity," one would have to say that this is the limit from the right. The limit from the left is "minus infinity," that is, unboundedly negative. This sort of statement, in terms of limits, is what was presumably meant by the incorrect statement above. This is extremely sketchy math, so just steer clear of division by 0.

*<math>\int_0^1\frac{1}{x} = \infty\ \ </math>  Infinity has a special meaning for integrals.

*<math>\int_0^1\frac{1}{x} = \infty\ \ </math>  Infinity has a special meaning for integrals.

*<math>\|\mathbb{Z}\| = \infty\ \ </math>  The cardinality of the integers is infinite.

*<math>\|\mathbb{Z}\| = \infty\ \ </math>  The cardinality of the integers is infinite.