When deriving the Catalan numbers using generating functions, eventually you reach the step:
$C(x) = 1 + xC(x)^2$
which means
$xC(x)^2 - C(x) + 1 = 0$
Which, through the quadratic formula, means
$C(x) = \frac{1 \pm \sqrt{1 - 4x}}{2x}$
But most derivations go forward with the assumption that the $\pm$ is actually a $-$ instead of a $+$. There should be two solutions but they move forward as if there is only one.
What is wrong with going forward with $+$?
This happens because if : $$C(x)=\frac{1+\sqrt{1-4x}}{2x}$$ then letting $x \to 0^{+}$ all the terms in the powers series (except $C_0$ ) vanish so :
$$C_0 =\lim_{x \to 0^{+}} C(x)=\lim_{x \to 0^{+}} \frac{1+\sqrt{1-4x}}{2x}=\infty$$
But $C_0=1$ so this is a contradiction .
But if we try with the other then , using l'Hospital's :
$$\lim_{x \to 0^{+}} \frac{1-\sqrt{1-4x}}{2x}=\lim_{x \to 0^{+}}\frac{-\frac{1}{2} \cdot (-4)}{2\sqrt{1-4x}}=1 $$ and this agrees with $C_0=1$ .