Let 'f' be differentiable at c. Let y = ax + b be the equation of the tangent line to the graph of 'f' at point [c,f(c)]. Prove that
Lim F(x) - (ax + b) = 0
x->c x - c

- anonymous

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- dan815

whats with the x-c on the bottom there

- anonymous

Treat that limit as a fraction
i.e x -c is the denominator

- dan815

is f and F the same function in the question

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## More answers

- anonymous

Ah sorry, i must have typed it up wrong ><
it's f(x) - (ax + b)

- dan815

so u know this tangent line passes through the point c,f(c)

- dan815

we have an indeterminate form 0/0 u can use lhopitals

- anonymous

okay

- dan815

or split into 2 cases, when f(x) degree <1 and f(x) degree>1

- dan815

|dw:1433324646253:dw|

- dan815

|dw:1433324690567:dw|

- anonymous

Ah okay

- dan815

|dw:1433324730086:dw|

- dan815

are you allowed to use L'hopitals though?

- anonymous

yeah, I was about to substitute the points on the tangent into the differentiated limit to solve for:
lim
x->c

- anonymous

yeah absolutely

- dan815

ah i see

- dan815

well here is the intuition for it

- dan815

when you are very close to the c

- dan815

you are travelling on your graph close to the tangent line,
where as in the denomiator u are travelling horizotally towards C

- dan815

so technically if u are going towards C on the tangent line itself the numerator shud be approaching 0 much faster than the denominator

- dan815

eh that doesnt sound very clear lol xD forget it, if its confusing u

- anonymous

yeah I know what you mean, I graphically picture this
Thanks for the help by the way ^^

- dan815

|dw:1433324933036:dw|

- anonymous

no no, I know what's going on. It's just this particular question, I haven't gotten much practice, since I like doing the easy L'hopital questions haha

- dan815

like u see the values on the graph are much closed to the tangent line as small distances away

- dan815

closer*

- dan815

at small distances

- dan815

because the smaller distances it gets the better a linear approximation is to this graph f(x)

- dan815

infact it gets infintessimally better than the horizotal way of approaching on the number line

- dan815

|dw:1433325051734:dw|

- dan815

there are excepts to this way of looking at it though, for example of the graph f(x) being a horizontal line too

- dan815

k well cheers

- amilapsn

use these:
\[F^/(c)=a=\lim_{x\rightarrow c }\frac{f(x)-f(c)}{x-c}\\\frac{y-F(c)}{x-c}=a\text{ which will be the equation of the tangent line..... }\]

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