# Little o notation trouble

1. Jul 13, 2010

### demonelite123

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. Jul 13, 2010

### Office_Shredder

Staff Emeritus
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

3. Jul 13, 2010

### hunt_mat

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?