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

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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

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whats with the x-c on the bottom there
Treat that limit as a fraction i.e x -c is the denominator
is f and F the same function in the question

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Ah sorry, i must have typed it up wrong >< it's f(x) - (ax + b)
so u know this tangent line passes through the point c,f(c)
we have an indeterminate form 0/0 u can use lhopitals
okay
or split into 2 cases, when f(x) degree <1 and f(x) degree>1
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Ah okay
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are you allowed to use L'hopitals though?
yeah, I was about to substitute the points on the tangent into the differentiated limit to solve for: lim x->c
yeah absolutely
ah i see
well here is the intuition for it
when you are very close to the c
you are travelling on your graph close to the tangent line, where as in the denomiator u are travelling horizotally towards C
so technically if u are going towards C on the tangent line itself the numerator shud be approaching 0 much faster than the denominator
eh that doesnt sound very clear lol xD forget it, if its confusing u
yeah I know what you mean, I graphically picture this Thanks for the help by the way ^^
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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
like u see the values on the graph are much closed to the tangent line as small distances away
closer*
at small distances
because the smaller distances it gets the better a linear approximation is to this graph f(x)
infact it gets infintessimally better than the horizotal way of approaching on the number line
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there are excepts to this way of looking at it though, for example of the graph f(x) being a horizontal line too
k well cheers
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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