Simplifying expressions when taking limit to infinity When taking the limit as $x$ approaches infinity of the function $[(x-2)^2+4x-x^2]/5$. How far should I simplify? I can simplify the part in parentheses, but simplifying further will get me $4/5$ and I'm not sure if that is right. Also, I can simplify the limit as $x$ approaches infinity of $(x-2)^2/x^2$ to $(x^2-4x+4)/x^2$ and get the limit of 1, but if I simplify it further to $-4x+4$ I will get a different limit. Is there something that dictates when I should exactly stop simplifying?  
 A: If you simplify correctly, you will always get the same limit.
$1$)
$$f(x)=\frac{(x-2)^2+4x-x^2}{5}=\frac{x^2-4x+4+4x-x^2}{5}=\frac{4}{5}$$
Because the function represents a straight line, $\displaystyle \lim_{x\to\infty}f(x)=\frac45$. 
$2$)
$$g(x)=\frac{(x-2)^2}{x^2}=\frac{x^2-4x+4}{x^2}=1-\frac{4}{x}+\frac{4}{x^2}$$
It's crystal clear that the limit $\displaystyle \lim_{x\to\infty}g(x)=1$. 
Furthermore, there is no way you can simplify $g(x)$ into $4x-4$.
A: You can stop simplifying when your expression is simple enough for you to see what the limit is. The amount of simplification does not change the limit. If it seems to do so, you have made a mistake.
In this particular case your first limit is $\frac45$ and the second one is 1.
The mistake is that you cannot simplify your second expression to $-4x+4$.
A: In mathematics, simplifying $X$ to $Y$ means finding an expression $Y$ whose value is equal to the value of $X$, but which is simpler. One of the most fundamental rules of mathematics is that if $X=Y$ then you can substitute $X$ for $Y$ or $Y$ for $X$ anywhere and be guaranteed to get the same result. The only exceptions relate to names. For example if in one paragraph I define $x$ to be something and in another I define it to be something else then obviously I can't substitute one for the other. In effect $x$ is being used as a name for different things in this case. (Similarly you wouldn't substitute one person called John with another called John.) But apart from this quibble the substitution rule holds everywhere. There simply is no such thing is mathematics as a simplification rule you can only use up to a point. Maybe that will make life easier for you.
You can't simplify $(x^2-4x+4)/x^2$ to $-4x+4$. But this isn't because of some weird rule about limits. It's because there's no rule about cancelling a term in the numberator using the denominator like this.
(A time when you shouldn't subsititute is when solving an equation like $X=Y$. You could substitute $X$ for $Y$ on the right hand side and get $X=X$. But that's a useless equation. Nonetheless, it's still correct so that's why I said "shouldn't" instead of "can't".)
