What is the limit (x->0) of (sin 3x)/x The answer is 3 but I dont know how to get it.

Expert Answers
lfryerda eNotes educator| Certified Educator

Evaluating this limit uses the fundamental trig limit `lim_{x->0}{sin x}/x` and a substitution.

`lim_{x->0}{sin 3x}/x`      multiply by 3/3

`=3 lim_{x->0} {sin 3x}/{3x}`    change variables let u=3x

`=3lim_{u->0} {sin u}/u`   now use limit



The limit is 3.

tonys538 | Student

The limit `lim_(x->0)sin (3x)/x` has to be determined.

If we substitute x = 0 in `sin (3x)/x` , the result obtained is of the form `0/0` . This is an indeterminate number and l"hospital's rule can be used in this case to arrive at the required limit.

Replace the numerator and denominator by their derivatives.

`(sin 3x)' = 3*cos (3x)`

`(x)' = 1`

This gives the limit:

`lim_(x->0) (3*cos (3x))/1`

Now cos 0 = 1. Substituting x = 0 in the expression obtained above gives:

`(3*1)/1 = 3`

The required limit `lim_(x->0)sin (3x)/x = 3`