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This may be braindead, but I'm trying!

If I have a function $f$ and that function is not defined at some x, then asking for the derivative of the function at $x$ makes no sense since there is no $f(x)$ at $x$.

But if I want to find a gradient for that function as close as possible to x, then how does that work? Isn't that the same as the derivative at x? It's like, I can do the same calculation but I have to disregard the result because I'm asking for something that doesn't exist.

For example, if $f(x)=\frac{1}{x−2}$, then $f(x)$ is not defined at $x=2$. So I can't find the derivative at that point since it doesn't exist, But the limit is 2. But the limit is the derivative, and the derivative doesn't exist! I'm confused.

I felt like I understood this but I woke up this morning with no idea. Last week I was happily finding the volume of cylindrical wedges, now I can't understand limits O_O

Please set me straight.

EDIT: I think my problem is the way I'm thinking about limits. It seems that there are two limits and I'm confusing them. The limit that $f(x)$ approaches and the limit that $x+h$ approaches. In the above example where $f(x)=\frac{1}{x-2}$, $2+h$ approaches $2$ and $f'(x)$ is undefined since the numerator contains a division by zero.
Is that my answer?

2nd EDIT: This is what I'm really asking: How do I find $lim_{x\to a}f'(x)$?

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    A typo: $f$ is not defined at $f(x)$? you meant $x$ instead of $f(x)$, most probably.2012-01-06
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    No I meant $f(x)$. For example, if $f(x) = \frac{1}{x-2}$, then $f(x)$ is not defined at $x=2$. So I can't find the derivative at that point since it doesn't exist, but....hmm, I may have just answered my own question. So the limit is 2. But the limit is the derivative, and the derivative doesn't exist! I'm confused.2012-01-06
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    Again the same confusion! Read your question again and you will see it. Anyways, if $f(x)$ is not defined at say $x=a$, then $f'(a)$ is also not defined.2012-01-06
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    Yeah you're right, I meant to say $f$ is not defined at $x$. :)2012-01-06
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    @Nikhil: Yup, but I think he's interested in the limiting values at "undefined points".2012-01-06
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    Using the language properly is an important step in avoiding confusion. The phrase "that function is not defined at $f(x)$" is non-standard, and must *never* be used. The phrase "that function is not defined at $x$" is OK, although I much prefer "that function is not defined at $x=a$" since it is useful, as much as possible, to use $x$ as the name of a variable. If would also be OK to say "$f(a)$ is not defined." The limit as $x$ approaches $2$ of $x$ is indeed $2$, but that says nothing about the derivative of a general function $f(x)$ at $x=2$.2012-01-06
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    As to your $2$nd edit, to find $\lim{x\to a}f'(x)$, we ordinarily find an expression for $f'(x)$ for $x\ne a$, and study what happens to $f'(x)$ as $x$ approaches $a$. The limit need not exist. This is already the case with $f(x)=|x|$. Here $f'(x)=1$ if $x>0$, and $f'(x)=-1$ if $x<0$, so $\lim_{x\to 0}f'(x)$ does not exist. Another wilder example is $f(x)=x\sin(1/x)$ if $x\ne 0$, $f(0)=0$. Here $f'(x)$ goes crazy as $x$ approaches $0$.2012-01-06

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