I have two functions. f: U->R is C^1 and f(0)=0. g: U->R defined with the piecewise function

g(x)= {f(x)/x if x not equal to 0; and lim (h->0) of f(h)/h if x=0}

I need to show that g(0) exists and that g is continuous (i.e. C^0).

So I can see that g(0) implies that x=0 which means I need to show that the lim (h->0) of f(h)/h exists. I can see how f(h)/h came about, but i'm not sure how to find that it exists or that g is continuous. Thanks