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Little o notation trouble

  1. Jul 13, 2010 #1
    for those who have calculus by apostol vol.1, i refer to page 288. i am looking at the first example where he proves that tanx = x + (1/3)x3 + o(x3). he showed that 1/cosx = 1 + (1/2)x2 + o(x2) and therefore tanx = sinx / cosx = (x - (1/6)x3 + o(x4))(1 + (1/2)x2 + o(x2)) and that should equal x + (1/3)x3 + o(x3).

    i multiplied it out and got x + (1/3)x3 + o(x3) - (1/12)x5 - o(x5) +o(x4) + o(x6) + o(x4)o(x2).

    my question is where did the rest of the o's go and how come you are only left with o(x3)? why not o(x4)?
  2. jcsd
  3. Jul 13, 2010 #2


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    o(x4) is also o(x3) (think about why)

    Also, o(x3)+o(x3) is o(x3)

    See if you can figure out why these two should be true, and how they solve your problem
  4. Jul 13, 2010 #3


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    Big oh and little oh notation can be thought of as the following if f(x)=O(x^2), then
    \lim_{x\rightarrow 0}\frac{f(x)}{x^{2}}=\textrm{constant}
    If f(x)=o(x^{2}) then:
    \lim_{x\rightarrow 0}\frac{f(x)}{x^{2}}=0
    Does this help?
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