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Unformatted text preview: L9 – Limits at Inﬁnity; Horizontal Asymptotes x2 x2 + 1 f (x) = What happens to f (x) as x gets larger in a positive or negative direction? x→∞ lim f (x) x→−∞ lim f (x) 1 Def – Let f be deﬁned on ( a, ∞ ) , then lim f (x) = L means that the values of
x→∞ f (x) can be made arbitrarily close to L by taking x suﬃciently large. f (x) as x Def – The line y = L is called a lim f (x) = L or lim f (x) = L .
x→∞ x→−∞ of the curve if either 2 How many horizontal asymptotes can a graph have? x→∞ lim tan−1 (x) = x→−∞ lim tan−1 (x) = 3 Thm – If r > 0 is a rational number, then 1 = xr 1 = xr x→∞ lim and x→−∞ lim 4 x→−∞ lim x4 − 1 = 1 − x3 x→∞ lim x2 + 2x = x3 + 3x x→−∞ lim 5x(2x − 1)(x − 1) = 6 − x2 − 2x3 5 Shortcut for rational functions p(x) where p(x) is of degree n and q (x) is of degree m, then q (x) If f (x) = If n < m If n > m If n = m 6 Find all vertical and horizontal asymptotes of f (x) = √ 3x . 4x2 − 8 7 Evaluate:
x→∞ lim x2 − 2x − x x→∞ lim sin(x) x 8 x→∞ lim 5 − x2 2 − x2 x→∞ lim (x2 − x3 ) 9 x→∞ lim e−x = x→∞ lim ex = x→−∞ lim e x = 1 x→∞ lim 2 = 2 + ex x→−∞ lim 2 2 + ex 10 Sketch by ﬁnding intercepts and limits: y = x(x + 2)2 (1 − x)3 . 11 ...
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 Fall '08
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 Calculus, Asymptotes, Limits

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