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Question
Evaluate the following limit :
`lim_(x -> sqrt(2)) [(x^2 + xsqrt(2) - 4)/(x^2 - 3xsqrt(2) + 4)]`
Solution
`lim_(x -> sqrt(2)) [(x^2 + xsqrt(2) - 4)/(x^2 - 3xsqrt(2) + 4)]`
Consider, `x^2 + xsqrt(2) - 4 = x^2 + sqrt(2) x - 4`
= `x^2 + 2sqrt(2)x - sqrt(2)x - 4`
= `x(x + 2sqrt(2)) - sqrt(2)(x + 2sqrt(2))`
= `(x + 2sqrt(2)) (x - sqrt(2))`
`x^2 - 3x sqrt(2) + 4 = x^2 - 3sqrt(2)x + 4`
= `x^2 - 2sqrt(2)x - sqrt(2)x + 4`
= `x(x - 2sqrt(2)) - sqrt(2)(x - 2sqrt(2))`
= `(x - 2sqrt(2)) (x - sqrt(2))`
Now, `lim_(x -> sqrt(2)) [(x^2 + xsqrt(2) - 4)/(x^2 - 3xsqrt(2) + 4)]`
= `lim_(x -> sqrt(2)) ((x + 2sqrt(2))(x - sqrt(2)))/((x - 2sqrt(2))(x - sqrt(2))`
= `lim_(x -> sqrt(2)) (x + 2sqrt(2))/(x - 2sqrt(2)) ...[(because x -> sqrt(2)"," therefore x ≠ sqrt(2)","),(therefore x - sqrt(2)≠ 0)]`
= `(lim_(x -> sqrt(2))(x + 2sqrt(2)))/(lim_(x -> sqrt(2))(x - 2sqrt(2))`
= `(sqrt(2) + 2sqrt(2))/(sqrt(2) - 2sqrt(2))`
= `(3sqrt(2))/(-sqrt(2))`
= – 3
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