Is Infinity =Undefined?

Let’s start with the equation $$y =\frac 1{(x-1)}$$. The positive and negative limit of $x$ at $1$ both approach $+∞$, but at $x = 1$, $y$ is undefined.

I know this is because the denominator of the equation resolves to $0$, but why does $y$ become undefined instead of $+∞$?

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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:

1. Is infinity a number?
2. 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:
$$\lim_{x\to1^-}\frac1{x-1}=-\infty\\\lim_{x\to1^+}\frac1{x-1}=+\infty$$

The limit is defined if and only if the two sided limits are equal. They are not, so the limit is undefined.