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