I must have missed this in class.
How do I show that this limit does not exist?
lim(x->2) |x^2-4|/(x-2)
How do I show that this limit does not exist?
lim(x->2) |x^2-4|/(x-2)
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|x² - 4| is a piecewise function, defined by where its inside changes signs.
(x + 2)(x - 2) = 0
x = -2, 2
Therefore, it is defined as follows:
|x² - 4| = { x² - 4, for x >= 2
............ { 4 - x², for -2 <= x <= 2
............ { x² - 4, for x < -2
The equal signs don't really matter where they're placed, so long as the function remains continuous.
Therefore, when you take this limit, you'll have to take the right hand limit and left hand limit and see if the pieces match up. You should be able to figure out what to do now.
(x + 2)(x - 2) = 0
x = -2, 2
Therefore, it is defined as follows:
|x² - 4| = { x² - 4, for x >= 2
............ { 4 - x², for -2 <= x <= 2
............ { x² - 4, for x < -2
The equal signs don't really matter where they're placed, so long as the function remains continuous.
Therefore, when you take this limit, you'll have to take the right hand limit and left hand limit and see if the pieces match up. You should be able to figure out what to do now.
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|x^2-4|/(x-2)
=|x-2|/(x-2) * |x+2|
Now,|x-2|/(x-2) is a signum function, which is discontinuous at x=2
=|x-2|/(x-2) * |x+2|
Now,|x-2|/(x-2) is a signum function, which is discontinuous at x=2