Prove that if $\lim_{x\to\infty}\,f(x)=L$ and $\lim_{x\to\infty}\,g(x)=K$ then \begin{equation} \lim_{x\to\infty}\,f(x)+g(x)=L+K. \end{equation}

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Lesson Parent: 
Function Limits IX: Horizontal Asymptotes And The Limit Of $f(x)$ As $x$ Tends To Infinity
Problem: 

Prove that if $\lim_{x\to\infty}\,f(x)=L$ and $\lim_{x\to\infty}\,g(x)=K$ then
\begin{equation}
\lim_{x\to\infty}\,f(x)+g(x)=L+K.
\end{equation}

Answer: 

It is true that if $\lim_{x\to\infty}\,f(x)=L$ and $\lim_{x\to\infty}\,g(x)=K$ then \begin{equation} \lim_{x\to\infty}\,f(x)+g(x)=L+K. \end{equation}

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Solution: 

Since we assume that

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and

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we can find $Y_1$ such that <Sign in to see all the formulas> and $Y_2$ 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>.

We have now shown that for every <Sign in to see all the formulas> there exists a $Y$ such that when <Sign in to see all the formulas> then <Sign in to see all the formulas>. Therefore,

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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 is used extensively throughout calculus proofs and is usually used without explicitly mentioning that it is being used.