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प्रश्न
Evaluate the following :
`lim_(x -> ∞) [("a"x^3 + "b"x^2 + "c"x + "d")/("e"x^3 + "f"x^2 + "g"x + "h")]`
उत्तर
`lim_(x -> ∞) [("a"x^3 + "b"x^2 + "c"x + "d")/("e"x^3 + "f"x^2 + "g"x + "h")]`
= `lim_(x -> ∞)[(("a"x^3 + "b"x^2 + "c"x + "d")/(x^3))/(("e"x^3 + "f"x^2 + "g"x + "h")/(x^3))] ...[("Divide numerator and"),("denominator by" x^3)]`
= `(lim_(x -> ∞)("a" + "b"/x + "c"/x^2 + "d"/x^3))/(lim_(x -> ∞)("e" + "f"/x + "g"/x^2 + "h"/x^3)`
= `(lim_(x -> ∞) "a" + lim_(x -> ∞) "b"/x + lim_(x -> ∞) "c"/x^2 + lim_(x -> ∞) "d"/x^3)/(lim_(x-> ∞)"e" + lim_(x -> ∞) "f"/x + lim_(x -> ∞) "g"/x^2 + lim_(x -> ∞) "h"/x^3)`
= `("a" + 0 + 0 + 0)/("e" + 0 + 0 + 0) ...[lim_(x -> ∞) 1/x^"k" = 0, "k" > 0]`
= `"a"/"e"`
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