What is the limit of #sin(6x)/x# as x approaches 0?

Question

23309 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-07-23T13:58:12

#lim_(xto0)sin(6x)/x=6#

Let ,
#L=lim_(xto0)sin(6x)/x=lim_(xto0)sin(6x)/(6x) xx 6#

Subst. #6x=theta=>xto 0,then , thetato0#

So.

#L=lim_(theta to 0) (sintheta)/theta xx 6=(1) xx 6=6#

Answer 2 · 2018-07-23T13:58:12

#6#

#\lim_{x\to 0}\frac{\sin(6x)}{x}#

#=\lim_{x\to 0}\frac{6\sin(6x)}{6x}#

#=6\lim_{x\to 0}\frac{\sin(6x)}{(6x)}#

#=6\cdot 1\quad (\because \lim_{t\to 0}\frac{\sin t}{t}=1)#

#=6#

Answer 3 · 2018-07-23T14:15:55

#lim_(x->0)(sin(6x))/x=6#

#lim_(x->0)(sin(6x))/x#

Let #y=6x#
#y/6=x#
As #x# approaches #0#, #y# also approaches #0#.

#therefore lim_(y->0)(sin(y))/(y/6)#
#=lim_(y->0)(6sin(y))/y#
#=6lim_(y->0)sin(y)/y#
#=6#

Note: #lim_(a->0)sin(a)/a=1# is a common limit and has been proven countless times.