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Question
Evaluate the following limits:
`lim_(x -> 1) (sqrt(x) - x^2)/(1 - sqrt(x))`
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Solution
`lim_(x -> 1) (sqrt(x) - x^2)/(1 - sqrt(x)) = lim_(x -> 1) (sqrt(x) - (sqrt(x)^4))/(1 - sqrt(x)`
= `lim_(x -> 1) (sqrt(x) (1 - (sqrt(x))^3))/(1 - sqrt(x))`
= `lim_(x -> 1) sqrt(x) (((sqrt(x))^3 - 1))/(sqrt(x) - 1)`
Put y = `sqrt(x)`
Where x → 1
We have y → `sqrt(1)` = 1
`lim_(x -> 1) (sqrt(x) - x^2)/(1 - sqrt(x)) = lim_(y -> 1) ((y^3 - 1))/(y - 1)`
= `(lim_(y -> 1) y) (lim_(y -> 1) (y^3 - 1^3)/(y - 1))`
= `1 xx 3(1)^(3 - 1)`
`lim_(x -> 1) (sqrt(x) - x^2)/(1 - sqrt(x))` = 3
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