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Answer on Calculus Question for ALYN SHAHIRAH BINTI ABD HALIM

Question #11468
Expert's answer
This limit does not exists, indeed, consider two sequences

xn = 10 +

yn = 10 - 1/n


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


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

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
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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