# Simple question about dividing by zero, $y=\frac{x}{x}$ when $x=0$

Is there a rule that says you have to simplify equations before evaluating them? Would $y=\frac{x}{x}$ at $x=0$ be $1$ or undefined, since without reducing it, you'd divide by $0$. I know the equation can simplify to $y=1$, but I thought simplification just made the equation easier to work and was not necessary. It still should yield the same results. I don't no if this is a bad example as the equation reduces to a constant, but the question would still hold up for any power of $x$ divide by $x$. For example, $y=\frac{x^2}{x}$ still has the same issue. At $x=0$ does it evaluate to $0$ or undefined. Sorry if this is a dumb question. Just one of my musings that I've wondered about.

• $x\rightarrow 0$ not $x=0$ Oct 8, 2015 at 20:10
• $y=\frac{x}{x}$ would be undefined at $x=0$...The graph would have a hole at $x=0$ and everywhere else $y=1$. Oct 8, 2015 at 20:10
• Hint: Simplifyng $y=\frac{x^2}{x}$ you have $y=x$ if $x\ne 0$ .... but, as a function this is different from $y=\frac{x^2}{x}$ . Oct 8, 2015 at 20:10
• As long as $x \neq 0$ right @EmilioNovati ? Oct 8, 2015 at 20:11
• @randomgirl if y=x/x has a whole in it @ x=0, then is y=x/x not the same as y=1? Oct 8, 2015 at 20:12

The proper term to use is "removable discontinuity". You're correct that the expression equals 1 in pretty much every case - still, 0 is not a part of that function's domain. The function you gave is equivalent to this: $$f(x)=1, x\neq0$$
I believe more precisely than the "division by zero" that you mention in the title, your question is about $$0/0$$. To study it, it is better to turn to the plane: rather than $$z=f(x)=x/x$$, let's consider $$z=g(x,y)=x/y$$. Then you can define different ways to reach $$0/0$$. If you follow the $$x=0$$ line with decreasing $$y$$, you'll get $$\lim_{y\rightarrow 0} g(0,y) = 0$$. If you follow the line $$x=y$$, then $$\lim_{y\rightarrow 0} g(y,y) = 1$$. And if you follow the curve $$y=x^2$$, you obtain $$\lim_{x\rightarrow 0^\pm} g(x,x^2) =\lim_{x\rightarrow 0^\pm} 1/x = \pm\infty$$ !
Thus the function $$f(x,y)=x/y$$ does not have a continuous extension in $$(0,0)$$. The ratio $$0/0$$ is really 'undefined' for this reason. But if you know that you'll be approaching $$0/0$$ along a definite path (e.g. $$y=x$$), it is valid to simplify as you will obtain a function which is the continuous extension of the one you started with.