lim as x approaches 0 of x(sin(1/x))

in the question, lim as x apporches 0, but in the equation it is 1/x
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The sine of any number is always between -1 and +1. So the leading factor is x rather than the sine. Therefore the product is governed by x and, since x is approaching zero, the product must also approach zero. The limit is zero.

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