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

1 Answer
Dec 12, 2014

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#