Epsilon-Delta Proof: Prove sqrt(x)=sqrt(a)

  • Thread starter Thread starter tvguide123
  • Start date Start date
  • Tags Tags
    Proof
tvguide123
Messages
9
Reaction score
0

Homework Statement


Let a rep. any real number greater than 0
Prove that the limit as x->a of sqrt(x) = sqrt(a)

I hav to prove the above equation using using an Epsilon-Delta proof but I am not sure how to start it off.

2. The attempt at a solution

I assumed that if 0<|x-a|<d
then |f(x) - f(a)|
= |sqrt(x) - sqrt(a)|

I am allowed to use basic manipulations of numbers that preserved the equation and also make helper assumption values for delta if needed as long as i account for them in my proof.

I've been stuck on this question for 3-1/2 hours now so I would really appreciate any help!
 
Physics news on Phys.org
Try multiplying |sqrt(x) - sqrt(a)| by |sqrt(x) + sqrt(a)| /|sqrt(x) + sqrt(a)|
 
Given epsilon>0, let delta = epsilon*sqrt(a), and remember that sqrt(x) + sqrt(a) >= sqrt(a) if x>=0.
 
Ah thanks a bunch guys, I couldn't figure out the first step for so long!

cheers :)
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
Back
Top