Question

What is the limit as x approaches 0 of `sin(3x)/(2x)`?

Answer 1

You could use L'Hospital's rule, which tells us that if you are taking a derivative and both the numerator and the denominator are either `0` or `+- oo`;

`lim f(x)/g(x) = lim (f'(x))/(g'(x))`

In this problem, both `sin(3x)` and `2x` are zero for `x=0`. Therefore L'Hospital's rule applies, and we can take the derivative of each.

`d/(dx) sin(3x) = 3cos(3x)`

`d/(dx) 2x = 2`

So, by L'Hospital's rule;

`lim_(x->0) sin(3x)/(2x) = lim_(x->0) (3cos(3x))/2 = 3/2`