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Question
Evaluate: `lim_(x -> sqrt(2)) (x^4 - 4)/(x^2 + 3sqrt(2x) - 8)`
Solution
Given that `lim_(x -> sqrt(2)) (x^4 - 4)/(x^2 + 3sqrt(2x) - 8)`
= `lim_(x -> sqrt(2)) ((x^2 - 2)(x^2 + 2))/(x^2 + 4sqrt(2x) - sqrt(2x) - 8)`
= `lim_(x -> sqrt(2)) ((x + sqrt(2))(x - sqrt(2))(x^2 + 2))/(x(x + 4sqrt(2)) - sqrt(2)(x + 4sqrt(2))`
= `lim_(x -> sqrt(2)) ((x + sqrt(2))(x - sqrt(2))(x^2 + 2))/((x + 4sqrt(2))(x - sqrt(2))`
= `lim_(x -> sqrt(2)) ((x + sqrt(2))(x^2 + 2))/(x + 4sqrt(2))`
Taking limits we have
= `((sqrt(2) + sqrt(2))(2 + 2))/(sqrt(2) + 4sqrt(2))`
= `(2sqrt(2) xx 4)/(5sqrt(2))`
= `8/5`.
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