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

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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