How to evaluate the limit <munder> <mo movablelimits="true" form="prefix">lim <mrow class

Karina Trujillo

Karina Trujillo

Answered question

2022-06-19

How to evaluate the limit lim x 0 sin 1 x x without using L Hospital

Answer & Explanation

seraphinod

seraphinod

Beginner2022-06-20Added 22 answers

We need to use here the Taylor series:
(1) arcsin ( x ) = ( x + x 3 6 + 3 x 5 40 + . . . . ) ,
Therefore,
lim x 0 arcsin x x = lim x 0 ( x + x 3 6 + 3 x 5 40 + . . . . ) x = = lim x 0 x ( 1 + x 2 6 + 3 x 4 40 + . . . . ) x = lim x 0 ( 1 + x 2 6 + 3 x 4 40 + . . . . ) = 1.
Here is the result:
lim x 0 arcsin x x = 1 .

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