Finding the limit of one function can be considered easy depending on the complexity of the function. But what happens when we want to combine two or more functions? Then some of the methods we used before can make finding the limit more complicated. In order to work with these types of limits in a more simplified way, we need to use different properties of limits. One of these properties is the property of taking the limity of the addition or substraction of two functions. We say that if lim f(x)=L as x→a   and   lim g(x)=M as x→a. Then  lim [f(x)-g(x)] =L-M as x→a

Claim:  lim [f(x)-g(x)] =L-M   as   x→a

Proof:

|f(x)-L|< ε/2 for  δ_1 and  |g(x)-M|<ε/2 for δ_2

We want to prove that lim [f(x)-g(x)] =L-M   as  x→a    i.e.   |f(x)-L|-|g(x)-M|<ε   for the smallest  δ between δ_1 and δ_2

|f(x)-L-[g(x)-M]|=|[f(x)-L]+[-g(x)+M]|

By triangle inequality |[f(x)-L]+[-g(x)+M]| ≤ |f(x)-L|+|-g(x)+M|

Note that |-g(x)+M|=|g(x)-M|

|f(x)-L- ( g(x)-M )|≤ |f(x)-L|+|g(x)-M|

We know that |f(x)-L|< ε/2  and  |g(x)-M|<ε/2

|f(x)-L- ( g(x)-M )|< ε/2 +ε/2

|f(x)-L- ( g(x)-M )|< ε