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Summer Bradford

Summer Bradford

Answered question

2022-06-13

How to prove that lim x 0 tan 2 x x 2 + x = 0
However, I'm not supposed to use L'Hopital's rule. I feel like squeeze theorem could be helpful but I can't find an adequate trigonometric property just yet. Any suggestions?

Answer & Explanation

stigliy0

stigliy0

Beginner2022-06-14Added 21 answers

Rewrite the function as ( tan x x ) 2 x x + 1
Bailee Short

Bailee Short

Beginner2022-06-15Added 3 answers

You have
(1) lim x 0 tan 2 x x 2 + x = lim x 0 sin 2 x cos 2 x ( x 2 ) ( 1 + 1 x ) = lim x 0 ( sin x x ) 2 ( 1 cos 2 x ) ( 1 1 + 1 x ) = 1 ( 1 ) ( 0 ) = 0
Note this uses the fairly well known lim x 0 ( sin x x ) = 1 limit (e.g., as shown in the Trigonometric functions section of Wikipedia's "List of limits" article).

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