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anonymous
 one year ago
If f is differentiable at 1, and the limit of f(1+h)/h as h>0 is 5, what is f(1)?
anonymous
 one year ago
If f is differentiable at 1, and the limit of f(1+h)/h as h>0 is 5, what is f(1)?

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jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1` f is differentiable at 1` so \[\Large \lim_{h\to 0}\frac{f(1+h)f(1)}{h}\] exists and it is defined based on the limit definition of the derivative

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so \[f'(1) = 5\lim_{h \rightarrow 0}\frac{f(1)}{h}\]

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1\[\Large \lim_{h\to 0}\frac{f(1+h)}{h}=5\] \[\Large \lim_{h\to 0}\frac{f(1+h){\LARGE \color{red}{0}}}{h}=5\] \[\Large \lim_{h\to 0}\frac{f(1+h){\LARGE \color{red}{f(1)}}}{h}=5\] so we see that \(\Large f(1) = 0\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so my step would be invalid?

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1it's valid, I just don't see where to go with it

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0oh ok. And you knew to put in that f(1) as 0 simply because the equation resembled the derivative limit?

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1I saw it match up with the limit definition. Well almost match up. It was just missing the `f(1)` part

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so f(x) = 0 and f'(x) = 5?

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1f ' (1) = 5 and f(1) = 0 we can't say anything about f(x) or f ' (x) in general

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1IF it was \[\Large \lim_{h\to 0}\frac{f(x+h)f(x)}{h}=5\] then we can say f ' (x) = 5
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