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Evaluate lim(xtooo) x^(3){sqrt(x^(2)+sqr...

Evaluate `lim_(xtooo) x^(3){sqrt(x^(2)+sqrt(1+x^(4)))-xsqrt(2)}.`

Text Solution

Verified by Experts

The correct Answer is:
`1//4sqrt(2)`
`underset(xtooo)limx^(3){sqrt(x^(2)+sqrt(1+x^(4)))-xsqrt(2)}`
`=underset(xtooo)limx^(3)[(x^(2)+sqrt(1+x^(4))-2x^(2))/(sqrt(x^(2)+sqrt(1+x^(4)))+xsqrt(2))]`
`=underset(xtooo)lim(x^(3))/((sqrt(x^(2)+sqrt(1+x^(4)))+xsqrt(2)))((1+x^(4)-x^(4)))/((sqrt(1+x^(4))+x^(2)))`
`=underset(xtooo)lim(x^(3))/((sqrt(x^(2)+sqrt(1+x^(4)))+xsqrt(2))(sqrt(1+x^(4))+x^(2)))`
`=underset(xtooo)lim(x^(3))/(x^(3)(sqrt(1+sqrt((1)/(x^(4))+1))+sqrt(2))(sqrt(1+(1)/(x^(4)))+1))`
`=(1)/(2sqrt(2)).1/2=(1)/(4sqrt(2))`
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