1.

Consider f(x)=(cos^(-1)(cos(sinx))-|x-pi|)/(sin^(3)x), then

Answer»

`underset(xto0)Limf(X)` =does not exist
f(x) has REMOVABLE discontinuity at `x=pi`.
JUMP of discontinuity at `x=pi" is "(1)/(3)`
f(x) is discontinuous at `x=npi,n in I`

Solution :`underset(xto0)Lim(COS^(-1){cos(sinx)}-|x-pi|)/(sin^(3)x)=underset(xto pi^(-))Lim(sinx+(x-pi))/(sin^(3)x)""(becausesinxgt0asxto pi^(-))`
Let `x-pi=t=underset(t TO0^(-1))Lim(-sint+t)/(-sin^(3)t)=(-1)/(6)`
R.H.L. `atx=pi`
`underset(x to pi^(+))Lim(-sinx-(x-pi))/(sin^(3)x)""(becausesingt0asxto pi^(+))`
Let `x-pi=t`
`underset(t to0)Lim(sint-t)/(sin^(3)t)=(1)/(6)`.


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