Please help

[kggirl]

f(x) = 3/2x – 5 , ﻉ = 0.001, x0 = 4 and L = 1, find a value of ä > 0 such that for all x satisfying |x - x0| < ä, the inequality |f(x) – L|< ﻉ holds.

I got stuck after getting 3.999 < x < 4.000. I was told the answer wasn't complete and should be 6.7x10^-4

the x0 is x sub zero

 

[soroban]

Hello, kggirl!

f(x) = 3/2x – 5 , ﻉ = 0.001, x₀ = 4 and L = 1.
Find a value of ä > 0 such that for all x satisfying |x - x0| < ä, the inequality |f(x) – L|< ﻉ holds.

I got stuck after getting 3.999 < x < 4.000.
I was told the answer wasn't complete and should be 6.7x10^-4

It looks like your work is correct . . . I think you simply went "too far".
. . You didn't have to solve for x . . . they wanted 'a'.

We are told: .|f(x) - L| .< .0.001

That is: .|(3/2)x - 5 - 1| .< .0.001 . ---> . |(3/2)x - 6} .< .0.001 . ---> . |(3/2)(x - 4)| < 0.001

And we have: .|x - 4| .< .(2/3)(0.001) .= .0.0006666.... . <--- There is our 'a' !

Therefore: .a .= .0.000666... .≈ .6.7 x 10^-4