Evaluate the following limit. DO NOT USE L'Hospital Rule?

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2 Answers
Jim S · Stefan V.
Dec 7, 2017

#lim_(xrarr0)sin(12x)/(3x) = 4#

Explanation:

Without using Rules De L'Hospital :)

#lim_(xrarr0)sin(12x)/(3x)#

  • #12x=u#

#x=u/12#

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

#=# #lim_(urarr0)sinu/(cancel(3)*u/cancel(12))# #=# #lim_(urarr0)sinu/(u/(4)# #=# #lim_(urarr0)(sinu/1)/(u/4)# #=#
#=# #lim_(urarr0)4sinu/u# #=# #4lim_(urarr0)sinu/u# #=# #4*1# #=# #4#

Monzur R.
Dec 7, 2017

A different way to find the limit

Explanation:

We shall use the well-known result #lim_(theta->0) sin theta/theta=1# and the limit constant multiple rule #limaf(x)=alimf(x)#

Also, notice #sin(12x)/(3x)=(4sin(12x))/(4(3x))=4*sin(12x)/(12x)#

#therefore lim_(x->0)sin(12x)/(3x)=4lim_(x->0)sin(12x)/(12x)=4*1=4#

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