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# True or false: if f'(r) exists, then lim x ->r f(x) = f(r)?

True or false

If f '(r) exists, then lim x->r f(x)=f(r)

Please give an explanation why.

Thanks!

### 1 Answer

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- 8 years agoFavorite Answer
assume r is any number.

{r ∈ ℝ}

as x becomes sufficiently close to that number, f(x) becomes arbitrarily close to f(r).

for example:

say r = 7

which means:

f(r) = f(7)

as x approaches r, x approaches 7.

when you've decided that x is sufficiently close to 7 to verify the limit to yourself, plug that x value (something damn near 7) into f(x) for x. you'll see that:

f(x) = f(7ish)

and remember that:

f(r) = f(7)

so by transitive property,

f(x) = f(r)

this is not a proof though. It's just an example to help verify the theorem.

so anyway,

the answer is True.

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