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Unformatted text preview: ) = ∞ • lim x → p 4 f ( x ) = 0. Note that if f and g have ∞ limits at p then we cannot say anything about the limit of f ( x ) g ( x ) and f ( x ) /g ( x ) there. (It can be basically anything or it might not exist.) Polynomials and rational functions It is not hard to show that for any nonconstant polynomial p ( x ) the limit of p ( x ) will be ∞ or∞ ad x → ∞ or x → ∞ . (Eventually the main term of the polynomial will dominate, so you will need to find the limit of cx n as x → ±∞ .) For rational functions the limits at infinity can also be a real number, this will happen if the degree for the numerator is at most as big as the degree of the denominator. Example 3. Show that lim x →∞ x 2 3 x +4 2 x 2 +4 x 5 = 1 2 . Solution. In order to find the limit at ∞ , it is enough to evaluate the function for large values of x . If x > 0 then x 2 3 x +4 2 x 2 +4 x 5 = 1 3 x + 4 x 2 2+ 4 x 5 x 2 whenever both sides make sense. But now the numerator has a limit equal to 1 as x → ∞ and the limit of the denominator as x → ∞ is 2 so the limit of the ration is 1 2 . One can also show that if the rational function f ( x ) = p ( x ) q ( x ) is not defined at x = c (because q ( c ) = 0) then lim x → c f ( x ) and lim x → c + f ( x ) will always exist (but might be infinite). Example 4. Show that lim x → 2 x 2 +2 x 2 =∞ and lim x → 2 + x 2 +2 x 2 = ∞ ....
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 Fall '08
 Staff
 Calculus, Limits, Limit, Fraction, Rational function, ∞

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