Relative Rates of Growth 8.3 Relative Rates of Growth

Quick Review

What you’ll learn about Comparing Rates of Growth Using L’Hôpital’s Rule to Compare Growth Rates Sequential versus Binary Search Essential Question How do we use calculus to understand growth rates as x→∞ and how it helps us understand the behavior of functions.

Faster, Slower, Same-rate Growth as x→∞ Let f (x) and g (x) be positive for x sufficiently large, 1. f grows faster than g (and g grows slower than f ) as x → ∞ if 2. f and g grow at the same rate as x → ∞ if

Example Comparing ex and x3 as x→∞ Show that e x grows faster than x 3 as x → ∞. The limit is of the indeterminate form ∞/ ∞, so we can apply L’Hôpital’s Rule.

Example Comparing ln x with x as x→∞ Show that ln x grows slower than x as x → ∞. The limit is of the indeterminate form ∞/ ∞, so we can apply L’Hôpital’s Rule.

Example Comparing x with x + sin x as x→∞ Show that x grows at the same rate as x + sin x as x → ∞. The limit is of the indeterminate form ∞/ ∞, so we can apply L’Hôpital’s Rule.

Transitivity of Growing Rates If f grows at the same rate as g as x → ∞ and g grows at the same rate as h as x → ∞, then f grows at the same rate as h as x → ∞.

Example Growing at the Same Rate as x→∞ Show that f and g grow at the same rate by showing that they both grow at the same rate as h(x) = x.

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Example Finding the Order of a Binary Search