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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?

Respuesta :

LammettHash
[tex]\displaystyle\lim_{x\to0}\frac{\tan x^2}x=\lim_{x\to0}x\frac{\tan x^2}{x^2}=\lim_{x\to0}\frac x{\cos x^2}\times\lim_{x\to0}\frac{\sin x^2}{x^2}[/tex]

Recall that

[tex]\displaystyle\lim_{x\to0}\frac{\sin x}x=1[/tex]

and so replacing [tex]x[/tex] with [tex]x^2[/tex], you get that the second limit is 1. Meanwhile,

[tex]\displaystyle\lim_{x\to0}\dfrac x{\cos x^2}=\dfrac01=0[/tex]

so the limit is 0.

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