Question

How do you prove that the limit of `x^(-1/2) = 2` as x approaches 1/4 using the epsilon delta proof?

Answer 1

Take for example `delta(epsilon)=min(1/8,3/32epsilon)`

Explanation:

Choosing `delta(epsilon)=min(1/8,3/32epsilon)`

Take `x : abs(x-1/4)< delta\ \ \ => 1/8< x< 3/8<1`

So

`abs(1/sqrt(x)-2)=abs((1-2sqrtx)/sqrtx)=abs((1-2sqrtx)/sqrtx*(1+2sqrtx)/(1+2sqrtx))=`
`=abs((1-4x)/(2x+sqrtx))`

But `sqrtx>x` when `x<1` so

`abs(1/sqrt(x)-2)< abs((1-4x)/(3x))< abs((4*(1/4-x))/(3/8))=32/3abs(x-1/4)<32/3delta<32/3*3/32epsilon=epsilon`