The value of $$2+\dfrac{1}{2+\dfrac{1}{2+\cdots}}$$ is ...
(a) $1-\sqrt{2}\quad$ (b) $1+\sqrt{2}\quad$ (c) $1\pm\sqrt{2}\quad$ (d) None of these
Now I have some question here from the way I saw it solved online.
He took $$2+ \frac1x = x$$
Now, my question is that why do we say $2+\frac1x$ equals to $x$? Like, it should be equal to $y$, right?
If we do that m it means we already know that RHS is $x$. I think that is wrong to do.
When we are not given like here nothing in RHS. Just "is" is written there. So what should we assume RHS as in these cases?
Answer for x = 1 + $\sqrt{2}$
Imagine that by some magic you have figured out the value of $x$. It's just a number. When you look at the first denominator in the expression you see exactly that expression again, so you can call it by the same name, $x$: $$ x = 2 + \frac{1}{ x} $$ and you can solve that equation for $x$.
The subtle problem with this argument is not that you used the same name $x$ twice, it's that you casually assumed that this expression that goes on forever actually makes sense. That assumption has to be justified - but not for this question. Since it's a multiple choice question the person who asked it is willing to assume the expression makes sense.
This kind of reasoning is common. For example, to express the number $$ x = 0.131313\ldots $$ as a fraction, you assume the infinite decimal makes sense. Then multiply by $100$ to see that $$ 100x = 13.1313\ldots = 13 + x. $$ Solving for $x$ yields $13/99$.
In each of these examples what's hidden is justifying algebra in what looks like an infinite expression.