Making a discontinuous function continuos?

Question

I have a piecewise function.

Need to find "B" so that F(x) is continuous

f(x)=
x^2 +b, x<2
x-b, x>or=2

Answer 1

lim -> 2+ = 2-b = f(2)

lim->2- = 4+ b
we need 4+b = 2-b
so b = -1

Answer 2

lim (x->2-) f(x) = lim(x->2-) x^2 + b = 4 + b
lim (x->2+) f(x) = lim(x->2+) x - b = 2 - b
To be continuous the left and right hand limits must be equal, so 4 + b = 2 - b. Hence b = -1.

Answer 3

to make the function continuous, you need that the two methods of calculating f(x) agree at their boundary, x =2/

so, 2^2+b = 2-b
2b = -2
b = -1