A question on Limits?

Question

Determine if the Limit

Lim
x->8
48 + abs (x-8) - 6x
---------------------------
7x - 56

exists, and if it does, compute its value.
1. limit = 1
2. limit does not exist
3. limit =5/7
4. limit = −5/7
5. limit = −1

Answer 1

do some factoring:

48 - 6x + abs(x-8)
------------------------ =
7x - 56

6(8-x) + abs(x-8)
----------------------- =
7(x-8)

all those x-8 will cancel out, but there are 2 cases. if x > 8, abs(x-8) = x-8, so you get

(-6 +1)/7 = -5/7.

at x=8, of course, the expression is undefined. if x < 8, abs(x-8) = -(x-8) = 8-x, so you get

(-6 - 1)/7 = -1.

Since lim x→8 from above is not equal to lim x→8 from below, the limit does not exist.

Answer 2

This would yield an indeterminant form so you can use L'Hopital's Rule. That means you can take the derivative of the top and bottom seperately until plugging in 8 yields an actual number

Answer 3

munkeyman below is right.
there is no general limit,
because coming from left gives you -1, and from right -5/7

Answer 4

does not exist. when graphed, there is an inconsistency in the graph. when x = 7.999 y = -1 and when x = 8.001 y = -.7143 that is too much of a gap to have an actual value. i say it Does Not Exist

Answer 5

I agree with munkeyman. out of the answers you have given.
you know it can be solved differently like mateo said but those choices aren't listed.

Answer 6

Could you please add somewhere what level mathematics this is.