1. Which of the following functions grow faster than e^x as x→...

Question

1. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?

a. x-3

Accepted Answer

Welcome back, everyone. Decide whether the function F of X equals X2d grows faster than eat to the power of X as X approaches infinity. For this problem, let's remember that to compare growth rates of two functions, we want to evaluate the limit as X approaches infinity of their ratio. We're going to use F of X in the numerator and G of X in the denominator. In this case, we can assign G of X to E to the power of X, so that we're evaluating the limit as X approaches infinity of X 2 divided by E to the power of X. And if we use direct substitution, we get infinity squared in the numerator, which is basically infinity, and eats the power of infinity in the denominator which is also infinity. And this indeterminate form tells us that we can apply Lato's rule. So let's go ahead and differentiate the numerator, the derivative of X 22 X. And differentiate the denominator. The derivative of eats the power of Xs eats the power of x. Now, let's substitute infinity for X again. We get to multiplied by infinity, that's infinity. And each to the power of infinity, which is infinity. So we still get the same in the terminate form. Which is suitable for lapel's rule to be applied once again. So let's differentiate. The numerator, the derivative of two axes 2. And again, the derivative of eats the power of Xs, eats the power of X. Once again, let's substitute infinity. We're going to get 2 divided by infinity. Which is going to give us a limit of 0, because we're dividing a constant by an infinitely large number. If the limit is 0, we have to recall that the denominator dominates over the numerator, meaning the denominator function. Essentially grows faster than the numerator, right? So in this case, we can say that eats the power of X grows faster. Because we got a limit of 0, or basically X squad. Grows Slower The E to the power of X. Let's label our final answer and thank you for watching.

1. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?a. x-3

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1. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?
a. x-3