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## Homework Statement

Prove that if all partial derivatives up to order [tex]n[/tex] are zero at [tex]\vec{x}[/tex] and [tex]f(x) = 0[/tex] then [tex]\displaystyle\lim_{h \rightarrow 0} \dfrac{f(x + h)}{|h|^n} = 0[/tex]

## Homework Equations

[tex]\displaystyle\lim_{h \rightarrow 0} \dfrac{f(x + h) - f(x)}{|h|} = 0[/tex]

[tex]f(x) = 0[/tex]

## The Attempt at a Solution

I am not sure if I should be doing this by induction. The first case is trivial. For the general case I am unable to move beyond either the limit [tex]\displaystyle\lim_{h \rightarrow 0} \dfrac{0}{|h|^{k}}[/tex]. I am also not sure how to formulate the equation for the k-th partial derivative. Is it permissible to use the same [tex]h[/tex] for both a given derivative and the next derivative? Thanks for your help.