Question 1139240: Find the indicated limit, if it exists. (2 points)
limit of f of x as x approaches 0 where f of x equals 5 x minus 8 when x is less than 0 and the absolute value of the quantity negative 4 minus x when x is greater than or equal to 0
-4
-12
-8
The limit does not exist.
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Piecwise Function:
Let
g(x) = 5x-8
h(x) = |4-x|
The f(x) function will change identity, so to speak, depending on what x is. There are two cases:
If  , then f(x) = g(x) = 5x-8
OR
If  , then f(x) = h(x) = |4-x|
Let's plug x = 0 into the g(x) function
g(x) = 5x-8
g(0) = 5*0-8
g(0) = 0-8
g(0) = -8
The input x = 0 leads to the output y = -8. We'll use this output later, so make sure to mark it.
Now plug x = 0 into h(x)
h(x) = |4-x|
h(0) = |4-0|
h(0) = |4|
h(0) = 4
We get an output of y = 4 here, in contrast to y = -8 earlier. Since these outputs do not match up (ie they are not equal), this means that we have a disconnect. The graph confirms this gap
Therefore, the limit ) does not exist.
More formally, the left hand limit (LHL) is -8 while the right hand limit (RHL) is 4. Since the LHL and the RHL aren't the same, this means the overall limit at this x value does not exist. For a limit to exist, the two sides must meet up at the same point (even if there is a hole at this meeting point).
Final Answer: The limit does not exist (choice D)
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