I was trying to find two limits which

Willie Kelley

Willie Kelley

Answered question

2022-04-07

I was trying to find two limits which are trivial using l'Hopital, but I can't seem to find a way to do it without it. For example :
f(x)=e2sin(x)1x
I tried to do it using the fact I know limex1x=1 as x0. Then,as 2sinx tends to 0 as well as x0, I first tried to prove the limit of f(x) would be 1, but I know it is 2 (using l'hôpital).

Answer & Explanation

Kapovkah7hpl

Kapovkah7hpl

Beginner2022-04-08Added 14 answers

limx0e2sinx1x=limx02sinxxe2sinx12sin(x)
=2limx0sinxxlimx0e2sinx12sin(x)
=2limx0sinxxlimy0ey1y
=2
aludirao09au

aludirao09au

Beginner2022-04-09Added 11 answers

For another approach, note that
e2sinx=1+2sinx+o(sin2x) so e2sin(x)1x=2sinx+o(sin2x)x2 as x0

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