Question

23. What roles do the functions e^x and ln(x) play in growth comparisons?

Accepted Answer

Understand that the function $$e^x$$ represents exponential growth, where the rate of increase is proportional to the current value, leading to very rapid growth as $$x$$ increases. Recognize that the function $$\ln(x)$$, the natural logarithm, grows much more slowly than any positive power of $$x$$ and certainly slower than exponential functions; it is the inverse of the exponential function $$e^x. $$When comparing growth rates, $$e^x$$ grows faster than any polynomial or logarithmic function, while $$\ln(x)$$ grows slower than any positive power of $$x. $$Use limits to compare growth rates formally, for example, evaluate $$\lim_{x 	o \infty} \frac{\ln(x)}{x^a} = 0$$ for any $$a \u003e 0$$, showing logarithmic growth is negligible compared to polynomial growth, and $$\lim_{x 	o \infty} \frac{x^a}{e^x} = 0$$, showing exponential growth dominates polynomial growth. Summarize that $$e^x$$ serves as a benchmark for rapid growth, while $$\ln(x)$$ serves as a benchmark for slow growth, making them fundamental in understanding and comparing different types of growth behaviors in calculus.

23. What roles do the functions e^x and ln(x) play in growth comparisons?

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Key Concepts

Exponential Function (e^x)

Natural Logarithm Function (ln(x))

Growth Rate Comparison