Question

Question regarding limits to see if my procedure of getting the answer is right!!!!

Limit x-> 0

Tan(x^2)
---------------
x

I know it's indeterminate so I need find another way


So I did the tan identity

cosx^2
----------
sinx^2
------------------------
x


cosx^2 1
------------ * ------
sinx^2 x


cosx^2
-----------
sinx^3


I don't know what to do, but I know that it has to be 0. Any ideas?

Answer 1

`\displaystyle\lim_(x\to0)\frac{\tan x^2}x=\lim_(x\to0)x(\tan x^2)/(x^2)=\lim_(x\to0)\frac x{\cos x^2}*\lim_(x\to0)(\sin x^2)/(x^2)`

Recall that


`\displaystyle\lim_(x\to0)\frac{\sin x}x=1`

and so replacing
`x` with
`x^2`, you get that the second limit is 1. Meanwhile,


`\displaystyle\lim_(x\to0)\frac x{\cos x^2}=\frac01=0`

so the limit is 0.