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Evaluate lim(xto0) ((1)/(x^(2))-(1)/(sin...

Evaluate `lim_(xto0) ((1)/(x^(2))-(1)/(sin^(2)x)).`

Text Solution

Verified by Experts

`underset(xto0)lim((1)/(x^(2))-(1)/(sin^(2)x))`
`=underset(xto0)lim((sin^(2)x-x^(2))/(x^(2)sin^(2)x))`
`=underset(xto0)lim((sinx+x)(sinx-x))/(x^(2)sin^(2)x)`
`=underset(xto0)lim(((x-(x^(3))/(3!)+(x^(5))/(5!)...)+x)((x-(x^(3))/(3!)+(x^(5))/(5!)...)-x))/(x^(2)(x-(x^(3))/(3!)+(x^(5))/(5!)...)^(2))`
`=underset(xto0)lim(-(2-(x^(2))/(3!)+(x^(4))/(5!)...)((1)/(3!)-(x^(2))/(5!)...))/(x^(2))=-1/3`
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