When a is not 0, f' tends to A. Because X tends to positive infinity, f(x) is infinite when X is positive infinity, so the ratio of f(x)/x infinity is not large.

Difference type, so the upper and lower derivatives become f', so LIMF (x)/x = LIMF' = A.

When a is 0, f(x) is always equal to a constant (set to c), then the original formula can be written as limC/x, and when x tends to infinity,

LimC/x tends to 0, that is, to a.

To sum up, the conclusion is proved.

Personal views for your reference.