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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 .
Dear Aditi,
Expand tan x using taylors theorem and then reduce the expression to get the answer.
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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
In case more explanation is needed, please email me at:
Dear Aditi
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
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