limit x tends to infinity ln((+/x))*-x  

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sciencesolve eNotes educator| Certified Educator

The request of the problem is vague, hence if you need to solve the limit `lim_(x->oo) (ln x)*(-x), ` hence, you need to substitute `oo`  for x in equation such that:

`lim_(x->oo) (ln x)*(-x) = (ln oo)*(-oo) = oo*(-oo)`

`lim_(x->oo) (ln x)*(-x) = -oo`

If you need to solve the limit `lim_(x->oo) (ln (x*(-x)), ` hence, you need to use logarithmic identity such that:

`ln(a*b) = ln a + ln b`

Reasoning by analogy yields:

`lim_(x->oo) (ln (x*(-x)) = lim_(x->oo) (ln x + ln(-x))`

Notice that the logarithm `ln(-x)`  is invalid for x>0, hence, the limit cannot be evaluated.

Hence, evaluating the limit `lim_(x->oo) (ln x)*(-x)`  yields `lim_(x->oo) (ln x)*(-x) = -oo.`