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Evaluate lim(xrarr1)(x^(2)-sqrtx)/(sqrtx...

Evaluate `lim_(xrarr1)(x^(2)-sqrtx)/(sqrtx-1).`

Text Solution

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`lim_(xto1)((x^(2)-sqrtx))/((sqrtx-1))=lim_(xto1){((x^(2)-sqrtx))/((sqrtx-1))xx((sqrtx+1))/((sqrtx+1))xx((x^(2)+sqrtx))/((x^(2)+sqrtx))}`
`lim_(xto1)((x^(4)-x)(sqrtx-1))/((x-1)(x^(2)+sqrtx))=lim_(xto1)(x(x^3-1)(sqrtx+1))/((x-1)(x^(2)+sqrtx))`
`=lim_(xto1)(x(x-1)(x^(2)+x+1)(sqrtx+1))/((x-1)(x^(2)+sqrtx))`
`=lim_(xto1)(x(x^(2)+x+1)(sqrtx+1))/((x^(2)+sqrtx))`
`=(1xx(1^(2)+1+)(sqrt1+1))/((1^(2)+sqrt1))=((1xx3xx2)/(2))=3["putting"x=1].`
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