1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Limit and partial derivatives proof

  1. Dec 19, 2007 #1
    1. The problem statement, all variables and given/known data

    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]

    2. Relevant equations

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

    3. 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.
  2. jcsd
  3. Dec 19, 2007 #2


    User Avatar
    Science Advisor

    Are you certain that the first case it "trivial"? Knowing that the partial derivatives are tells you that [tex]\lim_{h\rightarrow 0}\frac{f(\vec{x}+ h\vec{e_i})- f(\vec{x})}{h} = \lim_{h\rightarrow 0}\frac{f(\vec{x}+h\vec{e_i})}{h}= 0[/tex] where h is a real number and [itex]\vec{e_i}[/itex] is the unit vector in the ith coordinate direction. What you want to prove, [tex]\lim_{\vec{h}\rightarrow \vec{0}}\frac{f(\vec{x}+ \vec{h})}{|\vec{h}|^n}= 0[/tex] requires that h be a general vector going to the 0 vector. (It is, of course, very easy to prove one from the other. What is the derivative in direction [itex]\vec{h}[/itex] in terms of the gradient or partial derivatives?)
  4. Dec 19, 2007 #3
    Thank you for the reply. This was actually proved in our textbook, so I am just citing the result in the proof for this problem, leaving just the general case that needs to be proved.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook