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

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Lesson Parent:

Function Limits IV: The Gory Details Of One-Sided 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 $\lim_{x\to a^-}\,[f(x)+g(x)]=L+K$ if $\lim_{x\to a^-}\,f(x)=L$ and $\lim_{x\to a^-}\,g(x)=K.$

Solution:

Recall our definition of left-sided limits here. Since we assume that

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and

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

Now let's assume that we are at an $x$ such that <Sign in to see all the formulas>.

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we see that

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when <Sign in to see all the formulas>.

So we have shown that for every <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> such that if <Sign in to see all the formulas>, then <Sign in to see all the formulas>. Therefore,

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