Use the graph of f(x)=x2 below to find a number δ such that ∣∣x2−1∣∣<0.3 whenever 0<|x−1|<δ.

Question

https://prnt.sc/4Ui0oj7qDSOX

Answer 1

It helps to see that this is a conceptual question about limits, otherwise you're just doing mindless algebra and not using the graph.

In the picture, the blue parabola shows the graph of y = x² near the input value x = 1 and output value of y = 1. The inequality |x² - 1| < 0.3 represents the vertical range 0.7 < y < 1.3 (between the two horizontal red lines). You'll need to solve for the corresponding x values (x² = 0.7 gives x = 0.837 and x² = 1.3 gives x = 1.14).

The horizontal distances between x = 1 and the two x values found above are (1 - 0.837 = 0.163) and (1.14 - 1 = 0.14); you can choose δ to be any number less than both of these [although most places with list the "correct" answer as the smaller of the two distances, so 0.14 in this case].

Answer 2

-0.3 < x² - 1 < 0.3; 0.7 < x² < 1.3; √0.7 < x < √1.3; 1 - δ< x < 1 + δ

1 - δ = √0.7 or 1 + δ = √1.3; δ = 1 - √0.7 = 0.164 or δ = √1.3 - 1 = 0.141

We have to take smaller δ = √1.3 - 1 = 0.141