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Prove that if $\lim_{h\to 0}\,f(a+h)=L,$ then $\lim_{x\to a}\,f(x)=L.$

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Lesson Parent:

Function Limits III: The Gory Details

Problem:

Prove that if $\lim_{h\to 0}\,f(a+h)=L,$ then $\lim_{x\to a}\,f(x)=L.$

Answer:

It is true that $\mbox{if }\lim_{h\to 0}\,f(a+h)=L\mbox{ then }\lim_{x\to a}\,f(x)=L.$

Solution:

First, recall that

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means 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>. Also, recall that

<Sign in to see all the formulas>

means 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>.

Now let's say that $x-a=h$ . We know that

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since

$h$ is just a straight line function of

$x$ . Now when we are at a value of

$h$ such that when <Sign in to see all the formulas> then <Sign in to see all the formulas>.

Now, let's assume that

<Sign in to see all the formulas>

and that we are at an

$h$ such that <Sign in to see all the formulas>. We know by the definition of

<Sign in to see all the formulas>

that <Sign in to see all the formulas>. Since

$x-a=h$ , then <Sign in to see all the formulas> and therefore <Sign in to see all the formulas>. Since we already proved that <Sign in to see all the formulas> we know that when <Sign in to see all the formulas> then <Sign in to see all the formulas>. So by definition, this means that

<Sign in to see all the formulas>

and therefore

<Sign in to see all the formulas>