Suppose a state’s income tax code states that tax liability is 12% on the first $20,000 of taxable ...

lim_{x -> 0+} T(x)
= lim_{x -> 0+} a + 0.12x
= a,

so for this limit to be 0, we must have a = 0. For the limit as x approaches 20,000, there are two different pieces of T(x) on the left and right of 20,000, so the left and right limits must be computed separately:

lim_{x -> 20,000-} T(x)
= lim_{x -> 20,000-} a + 0.12x
= a + 0.12*20,000,

and

lim_{x -> 20,000+} T(x)
= lim_{x -> 20,000+} b + 0.16*(x - 20,000)
= b + 0.16*(20,000 - 20,000)
= b.

For lim_{x -> 20,000} T(x) to exist, the left and right limits must be equal, so

a + 0.12*20,000 = b.

Since a = 0, we have b = 0.12*20,000 = 2,400.

The fact that the first limit equals 0 means that people making very little money pay very small taxes. The existence of the second limit means that there is no jump in the tax at $20,000, i.e., people making a little bit less than $20,000 pay almost the same as people making a little bit more than $20,000.

The conditions on the limits are in a way not necessary to define T(x), as from the description of the tax code it immediately follows that a = 0, b = 2,400.