As noted above, the product rule cannot be used to find the limit of the given function because the limit of `e^(sin(pi/x))` does not exist.
But, the fact that the limit of `sqrt(x) e^(sin(pi/x))` is 0 can be proven using the Squeeze theorem, which says that if, for the functions f, g, and h such that
`f(x)<=g(x) <=h(x) ` in the vicinity of a
AND `lim_(x->a) f(x) = lim_(x->a) h(x) = L` (L being a finite number)
THEN `lim_(x->a) g(x)` also exists and equals L.
In our case, `sin(pi/x)` , while undefined at x = 0, is between -1 and 1:
Since `e^x` is an increasing function, the following also has to be true:
We can multiply all sides of this inequality by ` ` `sqrt(x)` , because it is always positive:
`sqrt(x) e^(-1)<=sqrt(x)e^(sin(pi/x)) <=sqrt(x)e^1`
This is true everywhere, including the vicinity of 0. So these three functions satisfy the conditions of the Squeeze theorem, because
`lim_(x->0+) sqrt(x)e^(-1) = lim_(x->0+) sqrt(x)e^1 = 0`
So the limit of the function "squeezed" in between is also 0:
`lim_(x->0+) sqrt(x)e^(sin(pi/x)) = 0`