A site called Dr. Math that had this answer. The short answer is "undefined", your first guess above. Here's the long answer: The set R of real numbers does not include an object called infinity, although when mathematicians work with the sets of cardinal or ordinal numbers there are objects that correspond to infinity in one of several senses.
The expressions 1/0 and 0/0 are not defined. That doesn't mean they are infinite. The standard definition for the set Q of all rational numbers is that Q is the set of all real numbers p/q, where p and q are integers and q is not zero. That leaves 1/0 and 0/0 not included in Q since they don't fit the definition.
Some people say that 1/0 is infinity as a kind of short hand for what happens to 1/x as x approaches 0. Note that if x approaches 0 from the right, 1/x becomes larger and larger; often we say that 1/x approaches infinity. Note also that 1/x becomes smaller and smaller as x approaches 0 from the left, so that 1/x approaches negative infinity. 0/0 is often used as shorthand for an indeterminate form in which numerator and denominator approach 0. The ratio is not determined in the sense that it can approach almost anything.
sin(x)/x is a 0/0 indeterminate form; as x approaches 0, both x and sin(x) approach 0; it is known that sin(x)/x approaches 1 as x approaches 0. Try calculating sin(x)/x for x = 0.01, 0.001, 0.0001, etc, in radians.
sqrt(|x|)/x is also a 0/0 indeterminate form; as x approaches 0, both sqrt(|x|) and x approach 0; it is known that the ratio becomes unbounded as x approaches 0. It is misleading to say that it approaches infinity since depending on whether x approaches 0 from the right or left, the ratio becomes very large or very small.
From The Math Forum.