Evaluate lim x → 0 tan ⁡ ( tan ⁡ x ) − sin ⁡...

Palmosigx

Palmosigx

Answered

2022-07-12

Evaluate lim x 0 tan ( tan x ) sin ( sin x ) tan x sin x

Answer & Explanation

sniokd

sniokd

Expert

2022-07-13Added 22 answers

lim x 0 tan ( tan x ) sin ( sin x ) tan x sin x = lim x 0 tan ( tan x ) tan ( sin x ) cos ( sin x ) tan x sin x = lim x 0 tan ( tan x ) tan ( sin x ) + tan ( sin x ) tan ( sin x ) cos ( sin x ) tan x sin x = lim x 0 ( tan ( tan x ) tan ( sin x ) tan x sin x + tan ( sin x ) ( 1 cos ( sin x ) ) tan x sin x )
Then, by mean value theorem, there is c ( π 2 , π 2 ) such that
tan ( tan x ) tan ( sin x ) tan x sin x = sec 2 c
and between sin x and tan x. Then lim x 0 tan ( tan x ) tan ( sin x ) tan x sin x = 1. Next, we will show that lim x 0 tan ( sin x ) ( 1 cos ( sin x ) ) tan x sin x = 1
lim x 0 tan ( sin x ) ( 1 cos ( sin x ) ) tan x sin x = lim x 0 tan ( sin x ) ( 1 cos ( sin x ) ) cos x sin x ( 1 cos x ) = lim x 0 tan ( sin x ) sin x 1 cos ( sin x ) sin 2 x sin 2 x 1 cos x cos x = lim x 0 tan ( sin x ) sin x lim x 0 1 cos ( sin x ) sin 2 x lim x 0 ( 1 + cos x ) lim x 0 cos x = 1 1 2 2 1 = 1.
Therefore,
lim x 0 tan ( tan x ) sin ( sin x ) tan x sin x = lim x 0 tan ( tan x ) tan ( sin x ) tan x sin x + lim x 0 tan ( sin x ) ( 1 cos ( sin x ) ) tan x sin x = 1 + 1 = 2

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?