Finding value of a limit

D

Denomination

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Sep 25, 2006
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Let f be the function defined by

f(x)= 5x+(x-4+|x-4|)^2

If the limit

lim f(8+h)-f(8) (the top is supposed to be over h as h goes to zero)
h->0 h

exsists,find it's value.

I know this is a long problem and I have work but I don't even know if it's right.
ok here is what I have.

5(8+h) +((8+h)-4+ |(8+h)-4|)^2 - ((5(8)+ (8-4+ |8-4|)^2)

My main problem is that I can't get passed this because of the absolute values and because I'm not sure if I'm supposed to foil the whole thing out. I know there should be an easier way then writing it all out, then canceling.
 
Hint for removing absolute-value bars: If x = 8, then what is x - 4? So what is |x - 4|?

Note: If h -> 0, then 8 + h will be very close to 8. In particular, (8 + h) - 4 should have the same sign as 8 - 4.

Eliz.
 
I just want to make sure I'm understanding this correctly. So then |x-4| should equal 0? Also does the second part mean that I should convert all my (8+h)-4 to just 8-4? I'm sorry but I have had a very shaky math background and I am very uncertain when I do math.
 
x = 2
x-4 = -2
|x-4| = 2

x = 3
x-4 = -1
|x-4| = 1

x = 4
x-4 = 0
|x-4| = 0

x = 5
x-4 = 1
|x-4| = 1

x = 6
x-4 = 2
|x-4| = 2

x = 7
x-4 = 3
|x-4| = 3

Are we seeing a pattern?

Note: You don't have to let things scare you. Think about them. Play with them a little. Figure it out.
 
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