# Question

Two functions f , g : R → R are equal up to nth order at if
lim h → o f(a + h) – g(a + h)/hn =0

(a). Show that f is differentiable at if and only if there is a function g of the form g(x) = a0 + a1 (x – a ) such that f and g and g are equal up to first order at a.

(b). if f1 (a),., f(n) (a) exist, show that f and the function g defined by

g(x) = f(i) (a)/ i! (x – a)i

(a). Show that f is differentiable at if and only if there is a function g of the form g(x) = a0 + a1 (x – a ) such that f and g and g are equal up to first order at a.

(b). if f1 (a),., f(n) (a) exist, show that f and the function g defined by

g(x) = f(i) (a)/ i! (x – a)i

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