Hi, is anyone out there who is good with this kind of math? I have this bonus question that I want to finish up but can't seem to figure it out.

I am supposed to find the limit of the following

lim x->0^+ sqrt(x+sqrt(x+sqrt+(x+...
0^+ is the right handed limit as x approaches 0.

My first instinct tells me that the limit should equal 0 just by substituting in 0, but I sort of want more of a detailed answer. Is this answer correct? If anyone can help that be great.

Thanks

The problem is only to determine y = √( x + √( x + √... ))
Check that y² = x + y,
Solve for y (assume x > 0, so is y),
Find your limit. And no, it is not zero.

Thanks! this helps helps me out. Using what you said I get that the limit to be equal to 1. Do you think you can help me understand it using epsilon-delta? If not its ok cuz you helped me get the answer, but just wanting to see why this works in detail.

In this case, things get simplified :

Once you have found that y = ( 1 + √(1+4x) ) / 2, using a quadratic equation, you can "smell" that the limit is 1, as x tends to zero (as you said).
So you must form the subtraction : |y - 1| = y - 1, because y > 1 for every x > 0,
y - 1 = (√(1+4x) - 1 ) / 2

Now, since (1+2x)² = 1 + 4x + 4x² > 1 + 4x, then 1+2x > √(1+4x) :
0 < y - 1 < ( 1 + 2x - 1 ) / 2
0 < y - 1 < x

Let ε > 0. Let δ = ε : for every x < δ, |y-1| < ε.