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Answer 1 · 2017-04-07T00:33:17

The limit diverges

Recall,

$lim_(x->a)[f(x)*g(x)]=lim_(x->a)f(x)*lim_(x->a)g(x)$

Apply and evaluate the limit from above,

$lim_(x->0)(lnx*cotx)=lim_(x->0+)(lnx)*lim_(x->0+)(cotx)$

Evaluate,

$lim_(x->0+)(lnx)=-oo$

$lim_(x->0+)(cotx)=oo$

Therefore,

$(-oo)*(oo)$

$-oo$

Apply and evaluate the limit below,

$lim_(x->0)(lnx*cotx)=lim_(x->0-)(lnx)*lim_(x->0-)(cotx)$

$lim_(x->0-)(lnx)=-oo$

$lim_(x->0-)(cotx)=-oo$

Evaluate,

$(-oo)*(-oo)$

$=oo$

Since the limit from the above tends towards $-oo$ and the limit from below is $oo$ . Therefore the limit diverges.