Find the limit: lim x?0 sin | x | / x

We need to find the limits to limx→0 sin | x | / x

Solution

We shall find the limit as x approaches 0 from the left and as x approaches 0 from the right.

For x < 0, | x | = – x
limx→0 – sin | x | / x
= limx→0 – sin (- x ) / x
= – limx→0 – sin ( x ) / x
= -1
For x > 0, | x | = x
limx→0 + sin | x | / x
= limx→0 + sin x / x
= 1
The two limits from the left and from the right are different, therefore the above limit does not exist.
limx→0 sin | x | / x does not exist

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