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Question
Evaluate the following limits:
`lim_(sqrt(x) -> 3) (x^2 - 81)/(sqrt(x) - 3)`
Solution
`lim_(sqrt(x) -> 3) (x^2 - 81)/(sqrt(x) - 3)`
Put `sqrt(x) - y`,
When `sqrt(x) -> 3`,
We have y → 3
`lim_(sqrt(x) -> 3) (x^2 - 81)/(sqrt(x) - 3) = lim_(sqrt(x) -> 3) (((sqrt(x)^2))^2 - 3^4)/(sqrt(x) - 3)`
= `lim_(sqrt(x) -> 3) ((sqrt(x))^4 - 3^4)/(sqrt(x) - 3)`
= `lim_(y -> 3) (y^4 - 3^4)/(y - 3)`
`lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "na"^("n" - 1)`
= `4(3)^(4 -1)`
= 4 × 33
`lim_(sqrt(x) -> 3) (x^2 - 81)/(sqrt(x) - 3)` = 4 × 27
= 108
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