What is the Limit of cos(3x)^(5/x) as x approaches 0?

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Answer 1 · 2018-05-28T20:10:42

#lim_(xto0)(cos(3x))^(5/x)=1#

#(cos(3x))^(5/x)=e^(ln(cos(3x))^(5/x))=e^((5ln(cos(3x)))/x#

#lim_(xto0)(5ln(cos(3x)))/x##=5lim_(xto0)(ln(cos(3x)))/x=_(DLH)^((0/0))#

#=5lim_(xto0)((cos(3x))'(3x)')/cos(3x)#

#=-15lim_(xto0)(sin(3x))/cos(3x)#

#=_(x->0,y->0)^(3x=y)#

#-15lim_(yto0)siny/cosy=lim_(yto0)tany=0#

#lim_(xto0)(cos(3x))^(5/x)=lim_(xto0)e^((5ln(cos(3x)))/x#

Substitute

#(5ln(cos(3x)))/x=u#

#x->0#
#u->0#

#=lim_(uto0)e^u=e^0=1#

graph{(cos(3x))^(5/x) [-15.69, 16.35, -7.79, 8.22]}