Lim_(x->0)(sin(\root(x))/(\root(sinx))) = Lim_(x->0) (sin(\root(x)) / (\root(sinx))) * (\root(x))/(\root(x))

Using lim_x->0 (sinx/x) = 1, we get

= Lim_(x->0) (sin(\root(x)) / (\root(x))

Now, if x is tending towards 0 from right side i.e., 0+ then the limit will be 1 using [ lim_x->0 (sinx/x) = 1] and if x is tending towards 0 from left side i.e., 0- then the limit is not defined as sin(\root(x)) will be not-defined.

So, the limit to the given function does not exist.

Hope it helps.