Let a function f(x) be differentiable at a point x = a and f'(a) = 1. Find the limit

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- Oct 1st 2013, 10:42 AMbobcantor1983Find limit of differentiable f(x)
Let a function f(x) be differentiable at a point x = a and f'(a) = 1. Find the limit

Attachment 29360 - Oct 1st 2013, 11:01 AMPlatoRe: Find limit of differentiable f(x)
- Oct 1st 2013, 11:33 AMSlipEternalRe: Find limit of differentiable f(x)
- Oct 1st 2013, 01:01 PMhughmistlerRe: Find limit of differentiable f(x)
I do not quite understand how you went from to and incorporated the term into the summation. Also how does the outlying disappear?

*Assuming that all of this can be reduced down to**would I now consider?*

So here if you look at individual parts of the series you have: since would approach 0 as n approaches infinity. It would follow similarly for the denominator and would cause the whole series to converge to 0?

How does the fact that f(x) is differentiable at a point x=a and f'(a) = 1 come into play? - Oct 1st 2013, 04:00 PMSlipEternalRe: Find limit of differentiable f(x)
Let's write out the summation :

So, how many times are you subtracting ? Since there are 100 terms, you are subtracting it 100 times.

The outlying does not disappear. When you divide by a fraction, that is the same as multiplying by its reciprocal. So, inside the summation we have:

Now, you can factor from every term, bringing it outside the summation (you can do this since does not depend on ).

Use your limit laws. The limit of a sum is the sum of the limits, so long as those limits exist (and so long as the sum is finite... infinite sums require more work). So:

As , . So, we can rewrite this limit:

Next, you asked about how you use . Use the definition of the derivative. For a general function , you know (Hint: When you write it out, replace by )

For your question about the series converging to 0, no. You have the numerator and denominator approaching zero. 0/0 is not defined, so you need to simplify the expression before you can take the limit. - Oct 1st 2013, 09:43 PMbobcantor1983Re: Find limit of differentiable f(x)
Ok I think I have this entirely put together, can you tell me if this looks right?

Now, since the limit of a sum is the sum of the limits, we can write:

As , . So, we can rewrite this limit:

Note: the definition of a derivative is

We have this here in our modified limit: .

So we can write

Is that straightforward and correct? - Oct 2nd 2013, 05:05 AMSlipEternalRe: Find limit of differentiable f(x)
Looks good to me