How do I find the value of $\sqrt{0+\sqrt{0+ \ ...}}$? I let $x= \sqrt{0+\sqrt{0+...}} \Rightarrow x = \sqrt{0+x}$
Solving for $x$ I get $x= 0$ or $x=1$.
I know the answer should be $x=0$ but why am I getting $1$ as solution in the first place? 1 also satisfies $x= \sqrt{0+x}$ but I need to eliminate one of the options as only one of it is correct but I can't seem to think of a correct reason to do so.
Any help?
PS : Please do not uses sequences in your explanation.
 A: You need to first define what you mean by $\sqrt{0+\sqrt{0+ \dots}}$
If you mean that 
$$\sqrt{0+\sqrt{0+ \ ...}} = \lim_{n\to\infty} \underbrace{\sqrt{0+\sqrt{\cdots \sqrt{0}}}}_{n\textrm{ times}}$$
then clearly, since all of the terms in the sequence you are calculating the limit of are $0$, the limit must also be $0$.

I see you don't want to use sequences, however, in that case, I need you to answer this question first:

How do you define $\sqrt{a_1+\sqrt{a_2+ \ ...}}$ for a given set of numbers $a_i$?

Unless you can answer this question, the expression $\sqrt{0+\sqrt{0+ \dots}}$ is more or less meaningless.
A: There is no reason to eliminate $x=1$, because it is a perfectly valid option.
Your nested radical has no meaning as long as you don't specify what $\cdots$ denotes. As it stands, your question is "what is the value of $\sqrt[2^n]\cdots$ ?"
Notice that 
$$x=\lim_{n\to\infty}\sqrt[2^n]a=1$$ for all $a>0.$
Actually, it is much more reasonable to consider the value $1$, as the set of initial values leading to $0$ has null measure, while the set leading to $1$ has probability $1$.
A: There are several answers and comments here that explain why this particular "calculation" fails. I think the question comes from a deeper misunderstanding. 
There are many places where a formal application of some "rule" produces wrong answers. The OP has asked about another, here: Powers of $i$ in Complex Numbers. . The explanation for the contradiction in each case depends on looking at the careful argument that leads to the rule, paying attention to the hypotheses in that argument that tell you when the rule is applicable.
Unfortunately, many students are taught that you do mathematics by applying rules. It takes practice to realize that the rules are codifications for underlying principles that should be understood. The real mathematics is in the understanding.
Typical situations that are prey to the problem of following a rule unthinkingly:


*

*dividing by a number that might be $0$

*working with  $\infty$ as if it were a number

*squaring "both sides of an equation"

*applying the "laws of exponents" to complex numbers

*using L'Hopital to find a limit

*manipulating an infinite sum (or other limit) as if it converged,
when it doesn't
