Problem

How does the long-term behavior of these functions compare: \(f(x)=x^3\) and \(g(x)=2^x\)?

Wide log-output comparison of x cubed and 2 to the x through x equals 18.

Big Picture

What this problem is really about

Long-run growth is about what persists after any early crossings, not which graph happens to be higher at one input. Compare how each output changes when the input increases by one: one model has a fixed multiplicative factor, while the other's factor approaches one. Tracking their ratio turns that structural difference into an eventual comparison.

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Preview only

As x gets large, neither function outgrows the other because they trade places and stay close for larger values of x.

As x gets large, f(x)=x³ and g(x)=2ˣ have the same long-term behavior because both functions keep increasing.

As x gets large, f(x)=x³ grows faster than g(x)=2ˣ because the exponent 3 makes the cubic increase more strongly.

As x gets large, g(x)=2ˣ eventually grows faster than f(x)=x³ because exponential growth outpaces polynomial growth.

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How does the long-term behavior of these functions compare: \(f(x)=x^3\) and \(g(x)=2^x\)?

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