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1. This is one of those probably really easy questions that I can't do for some reason... Probably lack of sleep. Can anyone help me? Most of the questions on the sheet are to do with the Mean Value Theorem, but I don't see how that applies...

Let f be continuous on (a, b) and be differentiable on each of (a, c), (c, b), where
a < c < b. Show that if f′(x) tends to L as x tends to c then f′(c) exists and equals L.
2. For any x in (c, b), the mean value theorem shows that

inf{f '(y) : y in (c, x)} <= [f(x) - f(c)] / (x - c) <= sup{f '(y) : y in (c, x)}

Letting x -> c from above, the inf and sup both tend to L, so

[f(x) - f(c)] / (x - c) -> L as x -> c from above

--

Similarly,

[f(x) - f(c)] / (x - c) -> L as x -> c from below
3. Thankyou!

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Updated: February 26, 2006
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