First of all, infinity is not a real number so actually dividing something by zero is undefined. In calculus $\infty$ is an informal notion of something "larger than any finite number", but it's not a well-defined number.
You might want to read the following:
- Is infinity a number?
- What exactly is infinity?
Edit: the following refers to an earlier version of the question.
Secondly, as the comments remarked, $-\infty\neq+\infty$ when talking about the real line. Note that when $x<1$ we have that $x-1<0$, and when $x>1$ we have $x-1>0$. Therefore:
The limit is defined if and only if the two sided limits are equal. They are not, so the limit is undefined.