## anonymous one year ago 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

1. dan815

whats with the x-c on the bottom there

2. anonymous

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

3. dan815

is f and F the same function in the question

4. anonymous

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

5. dan815

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

6. dan815

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

7. anonymous

okay

8. dan815

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

9. dan815

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10. dan815

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11. anonymous

Ah okay

12. dan815

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13. dan815

are you allowed to use L'hopitals though?

14. anonymous

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

15. anonymous

yeah absolutely

16. dan815

ah i see

17. dan815

well here is the intuition for it

18. dan815

when you are very close to the c

19. dan815

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

20. 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

21. dan815

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

22. anonymous

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

23. dan815

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24. 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

25. dan815

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

26. dan815

closer*

27. dan815

at small distances

28. dan815

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

29. dan815

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

30. dan815

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31. dan815

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

32. dan815

k well cheers

33. amilapsn

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