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I need help with this problem.

Let two real-valued functions $f,g$ be equal to the $n'$th order at $a$ if

$$\lim_{h\rightarrow 0}\frac{f(a+h)-g(a+h)}{h^n} = 0 $$

If $f'(a),\ldots,f^n(a)$ exist, show that $f$ and the function $g$ be given by

$$g(x) = \sum_{i=0}^n \frac{f^{(i)}(a)}{i!}(x-a)^i$$

are equal up to the $n$'th order at $a$.

I'm also given the following hint. The limit

$$\lim_{x\rightarrow a}\frac{f(x) -\sum_{i=0}^{n-1}\frac{f^{(i)}(a)}{i!}(x-a)^i}{(x-a)^n} $$

may be evaluated using L'Hospital's rule.

Can't I just plug in $g(a+h)$ and apply L'Hopital $n$ times? I'm not sure what to do with this hint.

2 Answers 2