how do you find sin7x/4x when lim x goes to 0

Question

the answer is 7/4

Answer 1

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.

Answer 2

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