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Evaluate: ("lim")(xvec2)(sqrt((x+7))-3sq...

Evaluate: `("lim")_(xvec2)(sqrt((x+7))-3sqrt((2x-3)))/(3sqrt(x+6) -2 3sqrt(3x-5))`

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We have L= `underset(xto2)lim(sqrt((x+7))-3sqrt((2x-3)))/(root(3)((x+6))-2root(3)((3x-5)))" "`(0/0 from)
Let `x-2=t` such that when `xto2,t to0.` Then
`L= underset(t to0)lim((t+9)^(1/2)-3(2t+1)^(1/3))/((t+8)^(1/3)-2(3t+1)^(1/3))" "`(0/0 from)
`=3/2underset(t to0)lim((1+t/9)^(1/2)-(2t+1)^(1/2))/((1+t/8)^(1/3)-(3t+1)^(1/3))" "`(0/0 from)
`=3/2underset(t to0)lim(1/2 t/9-(2t)1/2)/(t/8 1/3-(3t)1/3)=3/2(1/18-1)/(1/24-1)=34/23`
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