Use the given graph of f(x) = x^2 to find a number δ such that if |x-1|<δ then |x^2 -1|<1/2

Question

i just cant get the number of δ

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Answer 1

 

The black curve is f(x)=x². The red curve and pink shading is |x²-1|<½.

|x²-1|<½ when -√(3/2)<x<-√½ and √½<x<√(3/2) as (blue shading) shown  between the green and blue vertical lines. Plug examples into these inequalities: x=-1.2, -0.8, 0.8, 1.2, which satisfy the inequalities for x.

But we need values of δ > 0 which will give a value of x (where |x-1|<δ) in the shaded areas only. If, for example, δ=0.2, x-1<0.2, x<1.2; 1-x<0.2, x>0.8. Therefore, in this example, 0.8<x<1.2 when δ=0.2, which satisfies the inequality. In fact, in the extreme case, δ=√(3/2)-1, x-1<√(3/2)-1, x<√(3/2) and 1-x<√(3/2)-1, x>2-√(3/2).

So 2-√(3/2)<x<√(3/2), that is, 0.7753<x<1.2247.

So, at this point we have two values of δ which answer the question:

δ=0.2 and √(3/2)-1 (approx 0.2247).