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Calculus/calculating limits

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Expert: - 9/23/2009


Question
Hi. :) Could you tell me how to find the limit of absolute values?  (ex) /10-x/over 10-x when x is approaching 10? thanks in advance!

Answer
The problem is |10-x|/(10-x).

If x=10, the problem is undefined.

If x > 10, then 10-x<0, so |10-x| = x-10.  This makes the problem into (x-10)/(10-x) = -1.
So the limit(x->10 from the right) is -1.

If x < 10, the |10-x| = 10-x.  This makes the problem into (10-x)/(10-x) = 1.
So the limit(x->10 from the left) = 1.

Using both of these, it can be seen that function has no limit as x->10,
but only a left limit and a right limit.

Calculus

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