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Evaluate lim(x to 20) (x^(2)-4)/((sqrt(...

Evaluate `lim_(x to 20) (x^(2)-4)/((sqrt(3x)-2)-(sqrt(x)+2))`

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Given, `underset(xto20)"lim"(x^(2)-4)/((sqrt(3x)-2)-(sqrt(x)+2))= underset(xto2)"lim"((x^(2)-4)sqrt(3x-2)+sqrt(x+2))/((sqrt(3x-2)^(2)-sqrt(x+2))(sqrt(3x-2-sqrt(x+2))))`
`=underset(xto2)"lim"((x^(2)-4)(sqrt(3x-2)+sqrtx+2))/((3x-2)-(x+2))`
`=underset(xto2)"lim"((x^(2)-4)(sqrt(3x-2)+sqrt(x+2)))/(2x-4)`
`=underset(xto2)"lim"((x+2)(x-2)sqrt(3x-2)+sqrt(x+2))/(2(x-2))`
`=underset(xto2)"lim"(x+2)(sqrt(3x-2)+sqrt(x+2))/(2)`
`=(4(2+2))/(2)=8`
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