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Answer to Question #11468 in Calculus for ALYN SHAHIRAH BINTI ABD HALIM

Question #11468
lim┬(x→10)⁡〖|x-10|/(x-10)〗
Expert's answer
This limit does not exists, indeed, consider two sequences

xn = 10 +
1/n

yn = 10 - 1/n


Then

lim n->infinity
|xn-10|/(xn-10) = lim n->infinity |1/n|/(1/n) = 1

while

lim
n->infinity |yn-10|/(yn-10) = lim n->infinity |-1/n|/(-1/n) =
-1


On the other hand, there exist left and right limits, namely


left limit, i.e. when x->10 and x<10, and so xn-10<0, whence
|xn-10|=10-xn:
lim x->10-0 |xn-10|/(xn-10) = lim x->10-0
(10-xn)/(xn-10) = -1


right limit, i.e. when x->10 and x>10, and
so xn-10>0, whence |xn-10|=xn-10:
lim x->10+0 |xn-10|/(xn-10) = lim
x->10+0 (xn-10)/(xn-10) = 1

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