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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.