Leading Behavior Solutions

Leading Behavior If we consider a sum of functions in the limit that their arguments approach ∞, the behavior of the sum is dominated by the function that has the largest magnitude. We denote this term by f∞ , and say that f∞ is the leading behavior of f in the limit that x → ∞. f and f∞ have the same limit as x → ∞, so it follows that lim x→∞ f (x) =1 f∞ (x) 2 Example 1 Compare the rate at which f (x) = 2ex + ex + x12 and g(x) = 70ex + 12x2 approach ∞. Solution To compare the behavior of f and g as x → ∞ it is sufficient to compare f∞ and g∞ . Choosing the respective terms of f and g that approach ∞ at the fastest rate, we find 2 that f∞ (x) = 2ex and g∞ (x) = 70ex . Now we can compute the limit 2 f (x) 2 x(x−1) f∞ (x) 2ex = lim = lim = lim e =∞ lim x→∞ 70 x→∞ g∞ (x) x→∞ 70ex x→∞ g(x) Since f∞ (x) approaches ∞ faster than g∞ (x) (because the limit of the quotient approaches ∞), we can conclude that f (x) approaches ∞ faster than g(x). In fact, for large enough x, the leading behavior is a good approximation of f (x), so we can say that f∞ (x) 2 f (x) ≈ = ex(x−1) g(x) g∞ (x) 70 for large x. 1 Example 2 Compute the leading behavior of f (x) = e−x + x−2 + ln(x) as x → ∞. Solution Since all of the terms above are approaching 0 as x → ∞, we need to look for the term that approaches 0 the slowest (has the largest magnitude as x → ∞). Since all of the above terms are reciprocals of functions that approach ∞ as x → ∞, it follows that whichever denominator approaches ∞ the slowest will belong to the fraction that approaches 0 the slowest. Thus, we find that the leading behavior is given by 1 f∞ (x) = ln(x) 1 Example 3 Compute the leading behavior of f (x) = e−x + x−2 + ln(x) + 2 as x → ∞. Solution We now have a function where all of the terms except one are approaching 0 as x → ∞, so it follows that the term which is not vanishing, 2, is the leading behavior of f . (ie. f∞ (x) = 2). In an analagous fashion, we can consider the leading behavior of a sum of functions f as their arguments approach 0, which is given by the term that has the largest magnitude as x → 0. We denote this term by f0 . We find that f and f0 have the same limit as x → 0, so we find f (x) lim =1 x→0 f0 (x) We formalize this notion with the following definition Comparing the rate of growth of functions as x approaches 0 1. f (x) grows faster than g(x) as x → 0 if f (x) =∞ x→0 g(x) lim 2. f (x) grows slower than g(x) as x → 0 if f (x) =0 x→0 g(x) lim 3. f (x) changes at the same rate as g(x) as x → 0 if f (x) =L x→0 g(x) lim where L is any real number other than 0. Example 4 Compute the leading behavior of f (x) = x−1 + 2x−2 + x−3 as x → 0. Solution The larger the magnitude of the negative exponent is, the faster it approaches ∞ as x → 0. Thus, we find that f0 (x) = x−3 Example 5 Compute the leading behavior of f (x) = x + 2x2 + x3 as x → 0. Solution If we multiply two numbers less than 0 together, the result is an even smaller number. Thus, the highest power of x will approach 0 the fastest, so the lowest power will approach 0 the slowest. Thus, f0 (x) = x If we had a term such as x1/2 , it would approach 0 even more slowly so it would be the leading behavior. We can summarize these results into the following algorithm. Finding the Leading Behavior of a Function as x → ∞ and x → 0 Perform the following steps twice; once for x → ∞ and once for x → 0 1. If there are any terms that approach ∞, the dominant term is the one that approaches ∞ the fastest. If no terms approach ∞, proceed to step 2. 2. If there is a constant term, it is the dominant term. If there is no constant term, proceed to step 3. 3. The dominant term is the one that approaches 0 at the slowest rate. The dominant terms found for x → ∞ and x → 0 describe the leading behavior of the function. Using the leading behaviors of a function, we can construct a reasonable sketch of the function. A function f will behave similarly to f0 when x is near 0, and similarly to f∞ as x → ∞. By plotting the functions f0 and f∞ , we can sketch of f . Example 6 Use the method of leading behaviors to sketch f (x) = x + x1 for x ≥ 0. Solution When x is small, x1 is the dominant term, so f0 (x) = x1 . For large x, x dominates, so f∞ (x) = x. Now we graph these two functions, and sketch f (x) based on them. 10 8 x+x -1 6 4 2 0 0 2 4 6 8 10 x 2 Example 7 Use the method of leading behaviors to sketch f (x) = x2x 2 +4 for x ≥ 0. Solution When x is small, the denominator has leading behavior (x2 + 4)0 = 4, so f0 (x) = 2 2x2 = x2 . For large x, the denominator has leading behavior (x2 + 4)∞ = x2 , so f∞ (x) = 4 2x2 = 2. Now we graph these two functions, and sketch f (x) based on them. x2 3 2.5 1.5 2 2 2x (x + 4) -1 2 1 0.5 0 0 2 4 6 8 10 x Example 8 Use the method of leading behaviors to sketch f (x) = x22x+4 for x ≥ 0. Solution We know that (x2 + 4)0 = 4, so f0 (x) = 2x = x2 . Also, (x2 + 4)∞ = x2 , so 4 = x2 . Now we graph these two functions, and sketch f (x) based on them. f∞ (x) = 2x x2 1.4 1.2 0.8 2 2x (x + 4) -1 1 0.6 0.4 0.2 0 0 2 4 6 x 8 10