<s>[INST] Reduce a fraction Given the function 
$f(x)=\frac{x^2-5}{x+\sqrt{5}}$
If I draw this function in maple, I will get a line. How can that be true? I should expect a line except in area of $x = -\sqrt{5}$, where $f(x) \rightarrow \infty$ or $f(x) \rightarrow -\infty $. 
Of course Maple has factorized the numerator and reduced.
$\frac{x^2-5}{x+\sqrt{5}} = x-\sqrt{5},\ x\neq -\sqrt{5}$
My question is now, how can we ever reduce such a fraction with only condition $x\neq -\sqrt{5}$, when it is "$-\sqrt{5}$ and around it".
 [/INST] Note that $x^2 - 5 = x^2 - \sqrt 5^2 = (x-\sqrt5)(x+\sqrt 5)$, so we have
$$
f(x) = \frac{x^2 - 5}{x + \sqrt 5} = \frac{(x-\sqrt 5)(x + \sqrt 5)}{x+\sqrt 5} = \frac{x-\sqrt 5}{1} = x-\sqrt 5
$$
However, this manipulation does not suddenly make $f$ defined when $x = -\sqrt 5$. The only thing it does is that it simplifies the process of studying how $f$ behaves very close to $x = -\sqrt 5$, namely, it behaves exactly like the function $g(x) = x-\sqrt 5$, which is defined on the whole of the real line.
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