0

I know that the dividing by zero topic has been discussed in detail in this post. However, I still don't quite understand how $$\frac{1}{0}=undefined$$ but $$\lim_{n \to 0}\frac{1}{n}=\infty$$ isn't there some sort of contradiction?

Community
  • 1
qmd
  • 4,275
  • 1
    But using your notation, we have $\lim_{n\to 0}\frac{1}{-n} =-\infty$, so $1/0$ is both positive and negative infinity at the same time. If that doesn't smell "undefined", then I don't know what does. Plus, we don't like to have $\infty$ as an actual function value. We put it on limits more as a shorthand for " grows indefinitely ", rather than as an actual value. – Arthur May 19 '15 at 22:16
  • 1
    Note that in the function $$f(x)=\cases{1& if $x=0$\0& otherwise} $$ we have $\lim_{x\to0}f(x)=0$, because that's how limits are defined to work. – Arthur May 19 '15 at 22:24
  • Just use Real Projective Line and the two expressions become equivalent. – AlienRem May 19 '15 at 22:24

1 Answers1

1

There is only a contradiction if you misunderstand the limit $\lim_{x\to 0+}\frac {1}{x}$ to mean $\frac{1}{0}$. However, the correct meaning of limit is not just substitution, and no, there is no contradiction at all. You simply must understand that a limit (if it exists) has a precise meaning that has absolutely nothing to do (in this case) with division by $0$, only with divisions of the form $\frac{1}{x}$ for every possible positive value of $x$.

Ittay Weiss
  • 79,840
  • 7
  • 141
  • 236