Relative Rates of Growth Section 8.2

The exponential function grows so rapidly and the natural logarithm function grows so slowly that they set standards by which we can judge the growth of other functions... Comparing Rates of Growth

As an illustration of how rapidly grows, imagine graphing the function on a board with the axes labeled in centimeters… At x = 1 cm, the graph is cm high. At x = 6 cm, the graph is m high. At x = 10 cm, the graph is m high. At x = 24 cm, the graph is more than half way to the moon. At x = 43 cm, the graph is light-years high (well past Proxima Centauri, the nearest star to the Sun).

Let f (x) and g(x) be positive for x sufficiently large. Faster, Slower, Same-rate Growth as x  1. f grows faster than g (and g grows slower than f ) as if or, equivalently, if 2. f and g grow at the same rate as if (L finite and not zero)

According to these definitions, does not grow faster than as. The two functions grow at the same rate because Faster, Slower, Same-rate Growth as x  which is a finite nonzero limit. The reason for this apparent disregard of common sense is that we want “f grows faster than g” to mean that for large x-values, g is negligible in comparison to f.

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

Determine whether the given function grows faster than, at the same rate as, or slower than the exponential function as x approaches infinity. Guided Practice Our new rule: (because the base is less than one!) Grows slower than as

Determine whether the given function grows faster than, at the same rate as, or slower than the exponential function as x approaches infinity. Guided Practice Grows slower than as

Determine whether the given function grows faster than, at the same rate as, or slower than the squaring function as x approaches infinity. Guided Practice Grows faster than as

Determine whether the given function grows faster than, at the same rate as, or slower than the squaring function as x approaches infinity. Guided Practice Grows at the same rate as as

Determine whether the given function grows faster than, at the same rate as, or slower than the squaring function as x approaches infinity. Guided Practice Grows faster than as

Determine whether the given function grows faster than, at the same rate as, or slower than the natural logarithm function as x approaches infinity. Guided Practice Grows at the same rate as as

Show that the three functions grow at the same rate as x approaches infinity. Guided Practice

Compare the first and second functions: Rational Function Theorem! Compare the first and third functions: By transitivity, the second and third functions grow at the same rate, so all three functions grow at the same rate!