Where Do the Other o's Go in Little o Notation?

[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)?

 

[Office_Shredder]

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

 

[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?