Why is division by zero undefined?

For a nonzero divisor, writing $\frac{a}{b}=c$ is equivalent to saying $bc=a.$ For example, $12/3=4$ because $3\cdot4=12$. The result is also unique: if $3c=12$, then $c$ must be $4$.

That uniqueness is part of what makes division well-defined. Replacing $b$ with zero breaks exactly this property.

Suppose $1/0=c$. Reversing the operation would require $0\cdot c=1.$ But $0\cdot c=0$ for every real number $c$, so the equation is impossible. The same argument works for every $a\ne0$: the equation $0\cdot c=a$ has no solution. Therefore $a/0$ cannot be a real number when $a\ne0$.

If we try to set $0/0=c$, we get $0\cdot c=0.$ This equation is true for every $c$: $0$, $1$, $-7$, and every other real number all work. Division must return one unique quotient, so infinitely many possibilities do not define an answer. That is why $0/0$ is also undefined.

A limit studies values near a point; it does not perform the division at that point. For example, when $x\ne2$, $\frac{x^2-4}{x-2}=x+2,$ so $\lim_{x\to2}\frac{x^2-4}{x-2}=4.$ The original expression at $x=2$ is still $0/0$ and remains undefined.

Other behavior is possible: $1/x$ grows without bound positively as $x\to0^+$ and negatively as $x\to0^-$. It therefore has no single two-sided real limit. Using infinity to describe limiting behavior does not turn $1/0$ into a number.