# lim x→0   (etanx -  ex) / (tan x -x) is equal to?(A) 1 (B) 1/2 (C) 1/3 (D) 0[BOOK ANSWER: 1/2]EXPLAIN .

Aman Bansal
592 Points
12 years ago

Expand tan x using taylors theorem and then reduce the expression to get the answer.

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Aman Bansal

Chetan Mandayam Nayakar
312 Points
12 years ago

the Mcclaurin series expansion is

ey=∑y=0 to ∞yn/n!

therefore, lim x→0   (etanx -  ex) / (tan x -x)=1+(1/(tanx-x))(tan2x-x2+tan3x-x3...

=1+(1/(tanx-x))(tan2x/(1-tanx))-(x2/(1-x))=1+tanx+x=1

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jitender lakhanpal
62 Points
12 years ago

 lim x→0   (etanx -  ex) / (tan x -x)

take ex   as common

so we get

 lim x→0 ex  (etanx-x -  1) / (tan x -x) this is nothing but the standard limit limit x tends to 0 (ex - 1)/x = 1  where x = tanx - x  so it is equal to 1 so the answer is 1 and not 1/2    we can also check this by L hospital rule as it is 0/0