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

1 Answer
Nov 10, 2016

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