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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?

Verified Answer :

LammettHash

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

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,

\(\displaystyle\lim_{x\to0}\dfrac x{\cos x^2}=\dfrac01=0\)

so the limit is 0.

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