Question 1139240
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Piecwise Function:
<img src = "https://i.imgur.com/yIBizYh.png">
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 {{{x < 0}}}, then f(x) = g(x) = 5x-8
OR
If {{{x >= 0}}}, 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
<img src = "https://i.imgur.com/UrtPEN6.png">
Therefore, the limit *[Tex \Large \displaystyle \lim_{x\to 0}f(x)] 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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