solve the following limit without L'Hospital rule or series: <munder> <mo movablelimits="t

Sovardipk 2022-07-13 Answered
solve the following limit without L'Hospital rule or series:
lim x 0 arctan 2 x sin 3 x
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Answers (1)

Johnathan Morse
Answered 2022-07-14 Author has 18 answers
First you want to show that
lim y 0 arctan y y = 1
To do this, I would let θ = arctan y. Then θ 0 as y 0, so
lim y 0 arctan y y = lim θ 0 θ tan θ = lim θ 0 θ sin θ cos θ = 1 1 = 1
(the limit lim θ 0 sin θ θ = 1 is one of the basic limits you can usually invoke without proof).
Now
arctan 2 x sin 3 x = arctan 2 x 2 x sin 3 x 3 x 2 3
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