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?

e. (3/2)^x

Accepted Answer

Hello. In this video, we want to determine whether the given function f of x grows faster than the second function e x e to the power of x as x approaches infinity. So the problem gives us two functions to compare. The first function is F X equal to 5 divided by 4 raised to the power of x. And we are given a second function. Let's go ahead and name that G of X, but G of X is going to be e to the power of x. Now we want to determine if fx grows larger than g x when x approaches infinity. So in order to approach this problem, let's go ahead and use the limit comparison test. Now the limit comparison test is defined as the following. We are going to be observing a limit l, and this limit is going to be defined as the limit as x approaches infinity of fx divided by g of x. Now, these are going to be the conditions for the limit comparison test. If the limit is equal to 0. Then F of X grows slower than G of X. If the limit is greater than 0, then F of X grows at the same rate as G of X. And if the limit is pos is infinity. Then FX grows quicker than G of X. So these are going to be the conditions and the definition of the limit comparison test. So let's go ahead and set up the test with the 2 given functions. So we are going to define our limit L. L is going to be equal to the limit as X approaches infinity of F of X, which is 5/4 raised to the power of X, divided by G of X, which is E raised to the power of X. So what we need to do is we need to simplify. But notice that our numerator and denominator are raised to the same exponential value. What we can go ahead and do is rewrite the limit and raise the entire thing to one version of the exponential value so we can redefine the limit as the limit as x approaches infinity of the complex fraction 5/4 divided by e all raised to the power of x. And further simplifying the complex fraction leaves us with the limit as x approaches infinity of 5 divided by 4 e raised to the power of x. Now, if we are taking a look at the numerator and denominator, 4e has a greater value than 5. So what this tells us is that the absolute value of 5 divided by 4e is going to be bounded by 1, and the reason for this being is that since the numerator grows larger or since the denominator grows larger than the numerator, the value of 5 divided by 4e will never be larger than 1. It'll always be smaller than 1. Now we do have an identity for this. The identity states that if we have a limit as x approaches infinity of some exponential value 8 to the power of x and we know that this is equal to 0, this while this equals to zero if the upper bound of the base is going to be less than 1, which we do have in this condition. So what this tells us is that the limit. L, which is defined as the limit as x approaches infinity of 5 divided by 4 e raised the power of x is equal to zero. And since this is equal to 0, f grows slower. Than G of X by the limit comparison test. This is going to be the solution to the problem, and the overall solution is going to be option A. So I hope this video helps you in understanding on how to approach this problem, and we will go ahead and see you all in the next video.

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?e. (3/2)^x

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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?
e. (3/2)^x