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Evaluate lim(xtoa) (sqrt(a+2x)-sqrt(3x))...

Evaluate `lim_(xtoa) (sqrt(a+2x)-sqrt(3x))/(sqrt(3a+x)-2sqrt(x)),(ane0).`

Text Solution

Verified by Experts

We have,
`underset(xtoa)lim(sqrt(a+2x)-sqrt(3x))/(sqrt(3a+x)-2sqrt(x))" "("from"0/0)`
`=underset(xtoa)lim((sqrt(a+2x)-sqrt(3x))(sqrt(a+2x)+sqrt(3x)))/((sqrt(3a+x)-2sqrt(x))(sqrt(3a+x)+2sqrt(x)))`
`" "((sqrt(3a+x)+2sqrt(x)))/((sqrt(a+2x)+sqrt(3x)))" "("from"0/0)`
`=underset(xtoa)lim((a+2x-3x))/((3a+x-4x))((sqrt(3a+x)+2sqrt(x)))/((sqrt(a+2x)+sqrt(3x)))`
`=underset(xtoa)lim(sqrt(3a+x)+2sqrt(x))/(3(sqrt(a+2x)+sqrt(3x)))`
`=(sqrt(3a+x)+2sqrt(x))/(3(sqrt(a+2a)+sqrt(3a)))=1/3.(4sqrt(a))/(2sqrt(3a))=(2)/(3sqrt(3))`
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