Question

How do you find the limit of `(sin (2x)) / (sin (3x))` as x approaches 0?

Answer 1

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

Explanation:

This limit is indeterminate since direct substitution yields `0/0`, which means that we can apply L'Hospital's rule, which simply involves taking a derivative of the numerator and the denominator.

`lim_(x->0) sin(2x)/sin(3x) -> 0/0`, so applying L'Hospital's rule:

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

Graph of `sin(2x)/sin(3x)`:

graph{sin(2x)/sin(3x) [-3.015, 3.142, -0.607, 2.471]}

Answer 2

2/3

Explanation:

alternative answer

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

`lim_(x->0) 2* sin(2x)/(2x) *1/3 (3x) / sin(3x)`

`lim_(x->0) 2* sin(2x)/(2x) * lim_(x->0) 1/3 (3x) / sin(3x)`

let `u = 2x, v = 3x`

`lim_(u->0) 2* sinu/(u) * lim_(v->0) 1/3 (v) / sin v`

`= 2 * 1/3 = 2/3`