Proofs of limit properties

Hi when you are proving properties about limits of functions i.e the sum rule, the product rule and so on do you have to do seperate proofs for when.

the limits are at a finite point and when they are at infinity, but real valued?

Also when the limits are infintely valued do you need seperate proofs to show the properties in this case.

thanks

Can you give an example?

Strictly speaking you need different proofs for the +/-infinity cases. But the proofs are similar enough that people often don't worry about it.

so say f(x)->l as x->a and g(x)->p as x->a then the sum rule in this case would be involving real value limits at finite points.

f(x)+g(x) -> l + p as x-.a

So you could prove this if you wanted. This is the property given in begginers analysis books. But when i compute limits often i will be computing limits at infinity and limits which have infinite values. When i am computing limits at infinity but the limits involved have real values the sum rule above works.

say f(x)-> l as x-> infinity and g(x)->p as x-> infinity then

f(x)+g(x) -> l + p as x goes to infinity.

Now when you compute the limit you are really using the second property here i believe. Would this property require a seperate proof. thanks