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What is the limit as x approaches 1 when x-1 divided by absolute value of x-1?
Wiki User
∙ 14y ago
The limit doesn't exist. Consider approaching from the left (->) and the right (<-) [I'm assuming x is real, but if it is complex the same holds].
(->) : (x-1) / |x-1|. Since x<1, |x-1| = 1-x. So, we have
(x-1) / (1-x) = (x-1) / -1(x-1) = -1
(<-) : (x-1) / |x-1|. Since x>1, |x-1| = x-1. So, we have
(x-1) / (x-1) = 1
But recall that limits must be unique and clearly 1 doesn't equal -1. Thus, the limit doesn't exist.
Wiki User
∙ 14y ago
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