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Evaluate lim(x to oo ) [sqrt(a^(2)x^(2)+...

Evaluate `lim_(x to oo ) [sqrt(a^(2)x^(2)+ax+1)-sqrt(a^(2)x^(2)+1)].`

Text Solution

Verified by Experts

The correct Answer is:
`1//2`
`underset(xtooo)lim[sqrt(a^(2)x^(2)+ax+1)-sqrt(a^(2)x^(2)+1)]`
`underset(xtooo)lim(ax)/(sqrt(a^(2)x^(2)+ax+1)+sqrt(a^(2)x^(2)+1))" "`(Rationalizing)
`=underset(xtooo)lim(a)/(sqrt(a^(2)+(a)/(x)+(1)/(x^(2)))+sqrt(a^(2)+(1)/x^(2)))`
(Dividing numerator and denominator by x)
`=(a)/(2a)=1/2`
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