Question

Limit/Taylor series question:

Answer 1

\[\lim_{x \rightarrow 0}\large \frac{\cos(x)-1+\frac{x^{2}}{2}}{9x^{4}}\]

Answer 2

the first three terms of the taylor series of cos(x) are\[1-\frac{x^{2}}{2}+\frac{x^{4}}{24}\] so if you sub those in for cos(x) then everything cancels and you're left with \[\frac{1}{216}\]

Answer 3

looks good to me

Answer 4

my questions is why would i arbitrarily stop after the third term....if I continue with the taylor series then when i take the limit to zero i will be dividing by zero

Answer 5

the rest is automatically zero

Answer 6

for example next term is taylor series is has a \(x^6\) when you divide by \(x^4\) you get \(x^2\) and that clearly goes to zero

Answer 7

so all the rest of the terms have positive exponents

Answer 8

\[\Large \frac{-\frac{x^{6}}{24}}{9x^{4}}\] will be undefined at x=0

Answer 9

no, it will be \(-x^2\)

Answer 10

oooooooooooooooooooooooh.....i see....rookie mistake....cancel the x's and the limit wont be a problem

Answer 11

laws of exponents still work in this situation right?

Answer 12

right!!

Answer 13

thanks :)