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Question
Evaluate the following limits:
`lim_(x -> oo) (x^3 + x)/(x^4 - 3x^2 + 1)`
Solution
`lim_(x -> oo) (x^3 + x)/(x^4 - 3x^2 + 1) = lim_(x -> oo) (x^3(1 + x/x^3))/(x^4(1 - (3x^2)/(x^4) + 1/x^4)`
= `lim_(x -> oo) ((1 + 1/x^2))/(x(1 - 3/x^2+ 1/x^4)`
= `((1 + 1/oo))/(oo(1 - 1/oo + 1/oo))`
= `(1 +0)/(oo(1 - 0 + 0))`
= `1/oo`
`lim_(x -> oo) (x^3 + x)/(x^4 - 3x^2 + 1)` = 0
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