this means the answer will be between 1.9998 and 2.0002.
solve the equation for each of those and you should have your limits.
sqrt(1+x) = 1.9998
square both sides to get 1+x = (1.9998)^2
subtract 1 from both sides to get x = (1.9998)^2 - 1
solve for x to get x = 2.99920004.
next:
sqrt(1+x) = 2.0002
square both sides to get 1+x = (2.0002)^2
subtract 1 from both sides to get x = 2.0002^2 - 1
solve for x to get x = 3.00080004.
if x is between 2.99920004 and 3.00080004, then sqrt)(1+x) will be within 1.9998 and 2.0002.
this can be seen graphically.
the first graph is a far out look.
the values of x are so close together, that you can't distinguish between the 3 vertical lines.
the second graph is a near end look.
now you can see that the values of x shown lead to the values of y shown.
here's the graphs.
this could be modeled by an absolute value equation.
that equation would be:
|sqrt(1+x)-2| <= .0002
when the expression within the absolute value sign is positive, you get
sqrt(1+x) - 2 <= .0002
add 2 to both sides of the equation and you get sqrt(1+x) <= 2.0002.
when the expression within the absolute value signs is negative, you get -(sqrt(1+x) - 2) <= .0002
multiply both sides of that inequality by -1 and you get sqrt(1+x) - 2 >= -.0002
multiplying both sides of an inequality by a negative number reverses the inequality.
add 2 to both sides of that inequality to get sqrt(1+x) >= 1.9998
either way you look at it, your answer will be the same.
when x is equal to 2.99920004, it is within 3 - 2.99920004 = .00079996 units of 3.
when x is equal to 3.00080004, it is within 3.0080004 - 3 = .00080004 units of 3.
the differences from 3 are not the same.
to guarantee one value that will put the result within .0002 of 2, you would need to pick the smaller of the two results.
that would be .00079996 units.
since your answer doesn't require you to round any intermediate or final results, i would go with that.
note that the answer shown in the calculator is rounded to 8 decimal digits.
i had to go to excel to get more decimal digits.
the answer given by excel is:
calculator shows .00079996 while excel shows 0.0007999599999997110
calculator shows .00080004 while excel shows 0.0008000399999996690
the reason why the plus and minus units of x are not the same has to do with the squaring of both sides required to find the answer.