To compare exponential and quadratic growth as x approaches infinity, we see that exponential functions grow much faster than quadratic functions. Exponential functions start slowly but increase at an ever-accelerating rate, while quadratic functions grow at a steady, slower rate. Thus, the order of growth from fastest to slowest is: exponential functions, followed by polynomial functions (like quadratics), and then linear functions. Ultimately, exponential growth always surpasses quadratic growth at infinity.
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When we want to compare exponential and quadratic growth functions to see which grows faster as x gets larger, we look at how they behave as x approaches infinity. Exponential functions (like
or
) start off slow but grow much faster than quadratic functions (like
or
) when x becomes very large. This is because exponential functions increase at a rate that keeps getting bigger, while quadratic functions grow at a steady rate that doesn’t increase as quickly. So, when we arrange these functions from fastest to slowest growth as x increases, the order would be: exponential functions grow the fastest, followed by polynomial functions (like quadratic ones), and then linear functions (like
), which grow even slower. At infinity, exponential growth always outpaces quadratic growth.
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