Question

F(x) = e^x-1-x/x^2. ;x Not equal to 0. And 1/2 x=0

Answer 1

Answer:

Is |x|<1|x|<1?

Is −1<x<1−1<x<1? (x≠0x≠0)

So, the question asks whether x is in the range shown below:

Image

(1) x|x|<xx|x|<x

Two cases:

A. x<0x<0 --> x−x<xx−x<x --> −1<x−1<x. But remember that x<0x<0, so −1<x<0−1<x<0

B. x>0x>0 --> xx<xxx<x --> 1<x1<x.

Two ranges −1<x<0−1<x<0 or x>1x>1. Which says that xx either in the first range or in the second. Not sufficient to answer whether −1<x<1−1<x<1. (For instance xx can be −0.5−0.5 or 33)

Second approach: look at the fraction x|x|x|x| it can take only two values:

1 for x>0x>0 --> so we would have: 1<x1<x;

Or -1 for x<0x<0 --> so we would have: −1<x−1<x and as we considering the range for which x<0x<0 then completer range would be: −1<x<0−1<x<0.

The same two ranges: −1<x<0−1<x<0 or x>1x>1:

Image

(2) |x|>x|x|>x. Well this basically tells that xx is negative, as if x were positive or zero then |x||x| would be equal to xx. Only one range: x<0x<0, but still insufficient to say whether −1<x<1−1<x<1. (For instance xx can be −0.5−0.5 or −10−10)

Or consider two cases again:

x<0x<0--> −x>x−x>x--> x<0x<0.

x>0x>0 --> x>xx>x: never correct.

Image

(1)+(2) Intersection of the ranges from (1) and (2) is the range −1<x<0−1<x<0 (x<0x<0 (from 2) and −1<x<0−1<x<0 or x>1x>1 (from 1), hence −1<x<0−1<x<0):

Image

Every xx from this range is definitely in the range −1<x<1−1<x<1. So, we have a definite YES answer to the question. Sufficient.

Answer: C.