A Solution: Maths Fun Again.. Again

So the solution to my previous entry and the one prior to that was provided by a few folks as (root 2), which is actually quite trivial in my opinion. The actual question that has been bothering me was what i posted in the comments to whatever  few replies i got:

That is , what actually is . If the answer to both questions (blog entries) is , then how come is 2 as per my first blog entry and equal to 4 as per the second entry. This is the one I was actually looking for an answer to. I have kind of an idea what it could be, I will put my explanation here, and hopefully it makes some sense.

is actually limit of the sequence ,, ...

It is trivial to prove that the above sequence is increasing and is upper bounded by 2.

Coming back to the original question (the previous two blog entries marked above) which can be rewritten as .

 I tried to plot the graph of this function and it comes out as follows:

This proves that given a value of x, there are two values for y. And for x = , we know these values are 2 and 4. But we just indicated above that the sequence is upper bounded by 2. So the value 4 can't be possible for .

Hope it makes some sense.