hey, its been awhile since i could do these, could i have a little bit of help, if there is someone willing out there?

 

Cheers

 

Sarah

ok so far this is what i have gotten:

[math]

|\sqrt(x)-\sqrt(4)|=|\sqrt(x)-\sqrt(4)|\times\frac{|\sqrt(x)+\sqrt(4)|}{|\sqrt(x)+\sqrt(4)|}=\frac{x-4}{|\sqrt(x)+\sqrt(4)|}

[/math]

 

Let:

[math]

\frac{3}{2}<x<\frac{5}{2}

[/math]

 

Then

[math]

\frac{x-4}{|\sqrt(x)+\sqrt(4)|}\leq|x-4|\times\frac{1}{|\sqrt(\frac{3}{2})+2|}=|x-4|\times\frac{1}{\sqrt(\frac{3}{2})+2}

[/math]

 

So

[math]

|\sqrt(x)-\sqrt(4)|\leq|x-4|\times\frac{1}{\sqrt(\frac{3}{2})+2}

[/math]

 

Let

[math]

|x-4|\times\frac{1}{\sqrt(\frac{3}{2})+2}<\epsilon

[/math]

[math]

\therefore |x-4| < (\sqrt(\frac{3}{2})+2)\epsilon

[/math]

 

So choose

[math]

\delta = (\sqrt(\frac{3}{2})+2)\epsilon

[/math]

 

[math]

\therefore |x-4| < \delta \implies |h(x) - h(4)| < \epsilon

[/math]

 

well, yep, hows that?

...although i dunno how part 2 is going to work...

ok i've got another method....here we go

 

We want to find a [math] \delta > 0 [/math] (depending on [math] \epsilon [/math]) so [math] |x-4| < \delta \implies |h(x)-h(4)| < \epsilon [/math]

[math]

i.e. |\sqrt(x) - 2| < \epsilon

[/math]

 

Note [math] |\sqrt(x) - 2| = \frac{|x - 4|}{|\sqrt(x) + 2|} [/math]

 

Choose [math] \delta [/math] so that [math] \delta\frac{1}{|\sqrt(x) +2|} < \epsilon [/math]

 

Note [math] |x-4| < \delta \implies |x| < 4 + \delta [/math]

 

Choose [math]\delta[/math] to be < 1 then [math] |x| < 5 [/math]

 

Then choose [math] \delta = min(1,\epsilon(\sqrt(5) + 2)) [/math]

 

Then [math] |x-4| < \delta \implies |x|<=5, \ Since \ \delta<=1 [/math]

 

Then [math] |\sqrt(x) - 2| = \frac{|x - 4|}{|\sqrt(x) +2|} [/math]

 

[math] <\delta\frac{1}{\sqrt(5) + 2} <= \epsilon, \ since \ \delta <=\epsilon(\sqrt(5) +2) [/math]

 

Q.E.D

so what do peoples think?

lol, anyone at all?

Q.E.D

how was all of that related to Quantum ElectroDynamics?

how was all of that related to Quantum ElectroDynamics?

 

 

what are you talking about ? this is the maths problems section!

QED=Quantum ElectroDynamics, right?

oh ok , no i can't remember what is means, its a mathematical thing you put at the end of proofs, (like the little square boxes that are sometime used)

 

anyway, how was my proof (ignoring the Q.E.D bit )??

anyone? ... please :S ????

or how about this: (i think this is the one):

[math]

Want: \ |x - 4| < \delta \implies |\sqrt(x) - 2| < \epsilon

[/math]

[math]

x - 4 = (\sqrt(x) - 2)(\sqrt(x) + 2)

[/math]

[math]

|(\sqrt(x) - 2)(\sqrt(x) + 2)| < \delta

[/math]

[math]

x>=0

[/math]

[math]

\therefore \sqrt(x) + 2 >= 2

[/math]

[math]

\sqrt(x) - 2 < |(\sqrt(x) - 2)(\sqrt(x) + 2)|<\delta

[/math]

So Choose [math]\epsilon = \delta[/math]

QED is an abbreviation of Quod Erat Demonstrandum, which is Latin for "...which was to be demonstrated."

 

Shall look into the problem later today.

lol see i knew it meant something to do with proving!

 

thanks DQW

How do you get

 

[math]\frac{|x - 4|}{|\sqrt(x) +2|} ] <\delta\frac{1}{\sqrt(5) + 2}[/math] ?

don't worry about it, i never really understood, i'll just have to work it out some other time

 

thanks for the help though