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Question
Evaluate the following limits:
`lim_(x -> oo) ((2x^2 + 3)/(2x^2 + 5))^(8x^2 + 3)`
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Solution
`lim_(x -> oo) ((2x^2 + 3)/(2x^2 + 5))^(8x^2 + 3) = lim_(x -> oo)((2x^2 + 5 - 2)/(2x^2 + 5))^(8x^2 + 20 - 17)`
= `lim_(x -> oo) ((2x^2 - 5)/(2x^2 + 5) - 2/(2x^2 + 5))^(4(2x^2 + 5) - 17)`
= `lim_(x -> 00) (1 - 2/(2x^2 + 5))^(4(2x^2 + 5) - 17)`
Put 2x2 + 5 = y
When x → ∞
We have y = 2 × ∞ + 5 = ∞
x → ∞
⇒ y → ∞
∴ `lim_(x -> oo) ((2x^2 + 3)/(2x^2 + 5))^(8x^2 + 3) = lim_(y -> oo) (1 - 2/y)^(4y - 17)`
= `lim_(y - oo) (1 - 2/y)^(4y) xx (1 - 2/y)^(-17)`
= `lim_(y ->oo) (1 -2/y)^(4y) xx lim_(y -> oo) (1 - 2/y)^(- 17)`
= `(lim_(y -> oo) (1 - 2/y)^y)^4 xx (1 - 2/oo)^(- 17)` ........(1)
We know `lim_(x -> oo) (1 + "k/x)^x` = ek
(1) ⇒ `lim_(x -> oo) ((2x^2 + 3)/(2x^2 + 5))^(8x^2 + 3)`
= `(lim_(y -> oo)(1 + ((-2))/y)^y)^4 xx (1 - 0)^(- 17)`
= `("e"^(-2))^4 xx 1`
= `"e"^(-8)`
= `1/"e"^8`
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