Defining two limits

Question

Let limx→a(f(x)+g(x))=c1 and limx→a(f(x)−g(x))=c2, where c1 and c2 are constants. Find limx→a(f(x)⋅g(x)) and limx→a(f(x)/g(x)) and determine conditions of c1 and c2 so that such limits are well-defined

Answer 1

I read this as the limit as x→a of f(x)+g(x) is c₁ and f(x)-g(x) is c₂.

c₁+c₂=2f(x) and c₁-c₂=2g(x), therefore as x→a, limit of f(x)=½(c₁+c₂) and limit of g(x)=½(c₁-c₂). Therefore f(x).g(x)=¼(c₁²-c₂²) and f(x)/g(x)=(c₁+c₂)/(c₁-c₂) as x→a. To be meaningful, c₁≠c₂.

(EXAMPLE: Let f(x)=(3x-6)/(x²-4) and g(x)=(2x-4)/(x³-8) and a=2. c₁=11/12 and c₂=7/12. Now work out the required product and quotient of the functions with limits applied, and the expressions involving c₁ and c₂. You should find they match.)