Let f(x)={((alpha cotx)/x+beta/x^2 ,, 0<...

Let `f(x)={((alpha cotx)/x+beta/x^2 ,, 0<|x|lt=1),(1/3 ,, x=0):}`. If `f(x)` is continuous at `x=0` then the value of `alpha^2+beta^2` is

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The correct Answer is:

`underset(xrarr0)(lim)f(x)=(1)/(3)`
`rArr" "underset(xrarr0)(lim)(x.alphacotx+beta)/(x^(2))=(1)/(3)`
`rArr" "underset(xrarr0)(lim)(xalpha+betatanx)/(x^(2).tanx)=(1)/(3)`
`rArr" "underset(xrarr0)(lim)(alphax+beta(x+(x^(3))/(3)+...oo))/(x^(3)((tanx)/(x)))=(1)/(3)`
`rArr" "underset(xrarr0)(lim)((alpha+beta)x+((beta)/(3))x^(3)+...oo)/(x^(3))=(1)/(3)`
So, `" "alpha+beta=0`
Also,`" "beta=1rArr alpha=-1`

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