Verified answer

Since | x - a | approaches 0,

b - |x - a| ≤ f(x) ≤ b + |x - a|

approaches

b - 0 ≤ f(x) ≤ b + 0

b ≤ f(x) ≤ b

So, lim f(x) = b as x approaches a

If the limits of the quantities on the left and the right are equal (to L), then the limit of the one in the middle is also L.

Not really sure but I think the proper name is the 'mean value theorem'