Posted by Jin on Thursday, August 19, 2010 at 8:40pm.
One way would be to write sin 7x in infinite series form, and divide out 4x.
sin7x= 7x - (1/3!)(7x)^3 + (1/5!)(7x)^5 - ...
After dividing by 4x, the only term that does not vanish as x->0 is 7/4.
You could also use L'Hopital's rule. Take the ratio of the x-derivatives of the numerator and denominator.
Using L'Hospital rule
Lim sin7x/4x as x ->0 = Lim sim7x/Lim 4x as x->0 = Sin (Lim 7x)/(Lim 4x)as x-> 0 = 7/4 (on differentiating both numerator and denominator w.r.t. x) = 7/4
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