lecture15 - Math 006(Lecture 15 Infinite Limits Example 1...

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Unformatted text preview: Math 006 (Lecture 15) Infinite Limits Example 1. Let f (x) = 1 1 , g (x) = . Find lim f (x) and lim g (x). x→1 x→1 x−1 (x − 1)2 Definition 1. The vertical line x = a is a vertical asymptote for the graph of y = f (x) if f (x) → ∞, or f (x) → −∞, as x → a+ or x → a− . Definition 2. f (x) is called a rational function if f (x) = polynomials. Theorem 1. If f (x) = n(x)/d(x) is a rational function, d(c) = 0 but n(c) = 0, then the line x = c is a vertical asymptote of the graph of f . n(x) where n(x) and d(x) are d(x) 1 Example 2. Find the vertical asymptote(s) of the graph of y = f (x). (a) f (x) = (b) f (x) = (c) f (x) = (d) f (x) = x2 + x − 2 x2 − 1 x2 x−3 − 4x + 3 x2 + 20 5(x − 2)2 x−1 (x + 3)2 Limits at Infinity Example 3. Consider f (x) = 1 and g (x) = x2 . Find lim f (x) and lim g (x). x→∞ x→∞ x2 Theorem 2. If p is a positive real number and k is any real constant, then 1. lim k k = 0, lim p = 0 p x→−∞ x x→∞ x x→∞ x→−∞ 2. lim kxp = ±∞, lim kxp = ±∞ 3. If p(x) = an xn + an−1 xn−1 + · · · + a1 x + a0 , an = 0, n ≥ 1. then x→∞ lim p(x) = lim an xn = ±∞, x→∞ and x→−∞ lim p(x) = lim an xn = ±∞. x→−∞ Example 4. Let f (x) = 3x5 − 6x. Find lim f (x) and lim f (x). x→∞ x→−∞ 2 Finding Horizontal Asymptotes Example 5. Find lim 3x3 + 11 3x2 + x + 1 3x2 − 5x + 9 , lim and lim . x→−∞ x2 − 5 x→∞ x→∞ 2x2 − 5 x3 − 90 Theorem 3. Let f (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 , an = 0 and bm = 0. bm xm + bm−1 xm−1 + · · · + b1 x + b0 x→−∞ 1. If m = n, then lim f (x) = x→∞ lim f (x) = asymptote. x→∞ x→−∞ an an . The line y = is a horizontal bm bm 2. If n < m, then lim f (x) = lim f (x) = 0. The line y = 0 is a horizontal asymptote. 3. If n > m, then lim f (x) = lim f (x) = ±∞. f (x) does not have horizontal asympx→∞ x→−∞ tote Example 6. Find the horizontal asymptote(s) of the graph of y = f (x). 2x2 − 5 x2 + 4x + 4 x (b) f (x) = 2 x −4 (a) f (x) = (c) f (x) = x2 + 9 x−1 3 ...
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This note was uploaded on 09/04/2010 for the course MATH MATH006 taught by Professor Forgot during the Fall '09 term at HKUST.

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lecture15 - Math 006(Lecture 15 Infinite Limits Example 1...

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