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Evaluate underset(xto0)lim(5sinx-7sin2x+...

Evaluate `underset(xto0)lim(5sinx-7sin2x+3sin3x)/(x^(2)sinx).`

Text Solution

Verified by Experts

`underset(xto0)lim(5sinx-7sin2x+3sin3x)/(x^(2)xinx)`
`=underset(xto0)lim(5(x-(x^(3))/(3!)+...)-7(2x-(2x)^(3)/(3!)+...)+3(3x-(3x)^(3)/(3!)+...))/(x^(2)(x-(x^(3))/(3!)+...))`
`=underset(xto0)lim((-5x^(3))/(3!)+(56x^(3))/(3!)-(81x^(3))/(3!))/(x^(3)(1-(x^(2))/(3!)+...))`
`=(-5+56-81)/(3!)`
`=-5`
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