How do I evaluate the limit lim x → 0 ⌊ tan ⁡ x sin...

Desirae Washington

Desirae Washington

Answered

2022-07-13

How do I evaluate the limit
lim x 0 tan x sin x x 2

Answer & Explanation

Elijah Benjamin

Elijah Benjamin

Expert

2022-07-14Added 10 answers

Using Taylor–Young expansions: we know that
tan ( x ) = x + x 3 3 + o ( x 4 ) , sin ( x ) = x x 3 6 + o ( x 4 ) ,
hence
tan ( x ) sin ( x ) = x 2 + x 4 6 + o ( x 5 ) ,
and hence
tan ( x ) sin ( x ) x 2 = 1 + x 2 6 + o ( x 3 ) .
From here, we conclude that there exists a punctured neighborhood V of 0 such that
x V ,   tan ( x ) sin ( x ) x 2 > 1.
Since
lim x 0 tan ( x ) sin ( x ) x 2 = 1
we can assume that V has been chosen such that
x V ,   1 < tan ( x ) sin ( x ) x 2 < 2.
Hence
x V ,   tan ( x ) sin ( x ) x 2 = 1 ,
from which we conclude that
lim x 0 tan ( x ) sin ( x ) x 2 = 1.

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