I am looking for help with this question
By looking at the Taylor series, decide which of the following functions $\ln(1+y^2) \; \; \; \; \; \; \; \; \sin(y^2)\; \; \; \; \; \; \; \;1-\cos(y)$ is largest and which is smallest for values of $y$ near $0$.
I've done the Taylor expansions about $0$. I've written them below with the first few terms of the expansions.
$f(x)=\ln(1+x^2)$ $=x^2- \frac {x^4}{2}+\frac {x^6}{3}-\frac {x^8}{4}+\frac{x^{10}}{5}$
$--$
$f(x)=\sin(x^2)$ $=x^2- \frac {x^6}{6}+\frac {x^{10}}{120}-\frac {x^{14}}{5040}+\frac{x^{18}}{362880}$
$--$
$f(x)=1-\cos(x)$ $=\frac {x^2}{2}- \frac {x^4}{24}+\frac {x^6}{720}-\frac {x^8}{40320}+\frac{x^{10}}{3628800}$
How do I find which function is largest/smallest for values of $x$ near $0$? Each of these functions tends toward the point of origin. I'm having troubles grasping the intuition of the question.