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Question

If f(x)=x29x3+α,for x>3
= 5 , for x= 3
= 2x2+3x+β, for x< 3
is continuous at x= 3, find α and β.

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Solution

f(x)=x29x3+α,x>3
=5,x=3
=2x2+3x+β,x<3
And f(x) is continuous at x=3
For continuity, LHL and RHL at x=3 must be equal to f(3)
i.e., limx3f(x)=limx3+f(x)=f(3)
limh0f(3h)=limh0f(3+h)=f(3)
limh03h+9+α=limh0[2(3+h)2+3(3+h)+β]=5[3h<3&3+h>3]
3+9+α=2(9)+9+β=52
From equation 2,12+α=5
α=7
& 27+β=5
β=22

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