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what is the value of (cos(sinx)-cosx)/x4 as x tends to 0?

what is the value of (cos(sinx)-cosx)/x4  as x tends to 0?

Grade:Upto college level

1 Answers

Abhishek Doyal
askIITians Faculty 25 Points
10 years ago
When x tend to 0 its become 0/0
so by L hospital rule
f(x)=(cos(sinx)-cosx)/x4
so by L hospital rule
f(x)= - cosx.sin(sinx)+sinx/4x^3
again 0/0 form
L hospital rule
=sinx.sin(sinx)-cosx.cosx.cos(sinx)+cosx/12x^2
again 0/0 form
L hospital rule

cosx.sin(sinx)+sinx.cosx.cos(sinx)+sin2x.cos(sinx)+cos^3(x).sin (sinx)-sinx/24x
again 0/0 form
L hospital rule
= -sinx.sin(sinx)+cosx.cosx.cos(sinx)+cos2x.cos(sinx)-.5sin2x.cosx.sin(sinx)+2cos2x.cos(sinx)- sin2x.cosx.sin(sinx)-3cos^2(x)sinx.sin(sinx)+cos^3(x).cosx.cos(sinx)-cosx/24
put x=0
lim x tend to 0 equal to =1/6
Abhishek doyal
IIT Bombay
Askiitian tutor

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