reason for contrapositive

Let P be the statement of lim f(x) = a and lim g(x) = b, Q be the statement of lim (f(x)+g(x)) = a+b.

Obviously P → Q be the limit rule (can be proven by the strict definition of limits).

The statement: lim(f(x)+g(x)) not equal to a+b ≡ ¬Q,
lim f(x) not a or lim g(x) not b ≡ not [lim f(x) = a and lim g(x) = b] ≡ ¬P

∴ Its contrapositive: ¬Q → ¬P

From the first-order logic/ truth table, we get P → Q ⇔ ¬P ∨ Q
¬P ∨ Q ⇔ ¬P ∨ ¬(¬Q) ⇔ ¬(¬Q) ∨ ¬P ⇔ ¬Q → ¬P

Hence P → Q ⇔ ¬Q → ¬P, logically we explain the question.

if lim(f(x)+g(x)) not equal to a+b,

BUT lim f(x) = a

THEN

lim(g(x))
= lim(f(x) + g(x) - f(x))
= lim(f(x) + g(x)) - lim(f(x))
= a + b - a
= b

Therefore, the contrapositive is true.