Prove that \[\lim_{x\to a}\,[f(x)+g(x)]=L+K.\]

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Lesson Parent: 
Function Limits V: Properties of Function Limits
Problem: 

Prove that
\[
\lim_{x\to a}\,[f(x)+g(x)]=L+K
\]
if
\[
\lim_{x\to a}\,f(x)=L
\]
and
\[
\lim_{x\to a}\,g(x)=K.
\]

Answer: 

It is true that \[\lim_{x\to a}\,[f(x)+g(x)]=L+K.\]

SAMPLE SOLUTION

All of our problems have fully worked out, logical solutions. We connect every solution to its underlying lesson so you can understand the concepts needed to solve any problem and reinforce your learning by practicing with similar problems. All of the formulas are available to our subscribers, but here is a sample of the problem solution you will see when you log in:



Solution: 

Since we assume that

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and

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we can find <Sign in to see all the formulas> such that <Sign in to see all the formulas> and <Sign in to see all the formulas> such that <Sign in to see all the formulas>. Therefore we guess <Sign in to see all the formulas>.

Now because

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we see that

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when <Sign in to see all the formulas>.

As an aside, it is important to note that this means we can "distribute" the limit in like so:

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so that

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This fact that we can "distribute" the function limit like this will be used extensively throughout our calculus proofs. Sometimes this fact will be used without explicitly mentioning that it is being used so be on the lookout for its usage. Here is one example where it is used.