Proof by contradiction and division by $0$

Question

Proof by contradiction is based on the fact that if, as a consequence of a statement's truth, we reach a contradiction, then that statement must be false, since contradictions do not exist in mathematics.

So proof by contradiction assumes that there are no contradictions in mathematics. But a simple contradiction can be easily demonstrated by the fact that for some integer $x$, $\frac 10$ can either equal $\pm\infty$ as can be observed by plotting a graph for $\frac 1x$: at $x = 0$, the curve stretches to both infinities. This contradicts the fact that something cannot have two values at once, a core postulate of mathematics.

Is there a fallacy in my argument?

Answer 1

Your argument is based on a false assumption: far from having two values, $\frac10$ is undefined and therefore has no value. The statements $\lim\limits_{x\to 0^+}\frac1x=\infty$ and $\lim\limits_{x\to 0^-}\frac1x=-\infty$ are abbreviations for precise descriptions of how the function $f(x)=\frac1x$ behaves near (but not at) $x=0$; they say nothing about the undefined symbol $\frac10$.

Answer 2

I think the division of a nonzero number by $0$ is meaningless because if for example $a\neq 0$ and $a/0=b$ be a number so $0=0\times b=a$ and so we face to a simple contradiction. That's why we know $1/0$ as a meaningless thing. However $$\lim_{x\to 0^+}(1/x)=+\infty$$ shouldn't be mixed with that meaningless one.