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Evaluate: ("lim")(nvec0)(e^x-e^(xcosx))/...

Evaluate: `("lim")_(nvec0)(e^x-e^(xcosx))/((x+sinx)`

Text Solution

Verified by Experts

`underset(xto0)lim(e^(x)-e^(xcosx))/(x+sinx)=underset(xto0)lime^(x)(e^(x-cosx)-1)/(x+sinx)`
`=e^(0)underset(xto0)lim(e^(x-cosx)-1)/(x-xcosx).underset(xto0)lim(x-xcosx)/(x+sinx)`
`=1xx1xxunderset(xto0)lim(1-cosx)/(1+(sinx)/(x))`
=`(1-1)/(1+1)`
=0
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