If f(x)=underset(n to oo)lim(x^(n)g(x)+h...

If f(x)`=underset(n to oo)lim(x^(n)g(x)+h(x))/(x^(n)+1)`, show that,
f(x)`={{:(h(x)",","when" 0lt x lt 1),((1)/(2)[h(x)+g(x)]",", "when" x=1),(g(x)",","when "x gt 1):}`

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