theorem of l'hospital #lim x rarr +oo ((x^(-4/3))/(sin(1/x)))# ?

Question

#lim x rarr +oo ((x^(-4/3))/(sin(1/x)))#

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Answer 1 · 2018-05-06T20:34:42

The answer
#lim_(xrarr+oo)[4/(3*x^(1/3))]/[cos(1/x)]=(4/oo)/cos(0)=(4/oo)/1=0/1=0#

show below

#lim_(xrarr+oo)((1/x^(4/3))/(sin(1/x)))=0/0#

since the Direct compensation product equal #0/0# we will use l'hospital rule

#lim_(xrarra)f(x)/g(x)=0/0#

#lim_(xrarra)(f'(x))/(g'(x))#

#f(x)=x^(-4/3)#

#f'(x)=-4/3*x^(-7/3)#

#g(x)=sin(1/x)#

#g'(x)=cos(1/x)*-1/x^2#

#lim_(xrarr+oo)[-4/3*x^(-7/3)]/[cos(1/x)*-1/x^2]#

#lim_(xrarr+oo)[4/3*x^(-1/3)]/[cos(1/x)]#

#lim_(xrarr+oo)[4/(3*x^(1/3))]/[cos(1/x)]=(4/oo)/cos(0)=(4/oo)/1=0/1=0#

Note that:

#c/oo=0#

where c=constant