Video: Comparing Rates of Growth of Functions

If lim_(π‘₯ β†’ ∞) 𝑓(π‘₯)/𝑔(π‘₯) = ∞, what do you notice about the growth rate of 𝑓(π‘₯) compared to 𝑔(π‘₯) as π‘₯ β†’ ∞?

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Video Transcript

If the limit of 𝑓 of π‘₯ over 𝑔 of π‘₯ as π‘₯ approaches ∞ is ∞, what do you notice about the growth rate of 𝑓 of π‘₯ compared to 𝑔 of π‘₯ as π‘₯ approaches ∞?

This limit tells us that the quotient 𝑓 of π‘₯ over 𝑔 of π‘₯ grows without bound as π‘₯ approaches ∞. And for the quotient 𝑓 of π‘₯ over 𝑔 of π‘₯ to grow without bound, the function 𝑓 of π‘₯ must be growing faster than 𝑔 of π‘₯. Our conclusion is, therefore, that the growth rate of 𝑓 of π‘₯ is greater than that of 𝑔 of π‘₯ as π‘₯ approaches ∞. Another way of saying this is that the function 𝑓 of π‘₯ dominates the function 𝑔 of π‘₯.

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