Exponential growth rates
a. For what values of b &g...

Question

Exponential growth rates

a. For what values of b > 0 does bˣ grow faster than eˣ as x→∞?

Accepted Answer

Considering the functions F of X equals A X and G of X equals E X, for a greater than 0, when does F of X grow at a slower rate compared to G of X as X approaches infinity? We have 4 possible answers, being a greater than E. A is between 0 and E, when A equals E, or when A is between 0 and 2 E. Now, to solve this, we're going to make a ratio between our two functions. But before we do that, we need to rewrite our A to the X. We know that A to the X is equivalent. Uh Raised to the X natural log of A. Meaning then We have F of X equals E rates to the X natural log A. G of X equals the 2 E rates to the X. Now, let's examine the ratio. FX divided by G of X. We have E to the X natural log A divided by 2 E raised to the X. We can simplify this. To be 1/2 E base to be X multiplied by natural log of A minus 1. So now, we have our ratio. Let's check out the behavior of the limit. We have the limit As x approaches infinity. Of our function 1/2 E raised to the X, natural A minus 1. Now, we noticed that our limit value is dependent on whatever this natural log a minus 1's sign is. So let's check this. Natural log a minus 1. We will first check if it's less than 0. We get natural log of A is less than 1. And in this case, A would be less than E to the 1, or A is less than E. So when A is less than E, Our function Grows slower In other words, F of X. Grows slower. The GFX. And we can tell that based on our fraction. Now let's check if this expression. is greater than 0. We would get Nao A. Rareer than one Or A is greater than E. Now, when A is greater than E, if we look at our ratio again. Our function F of X. Would grow faster Finally, we can check. When natural like A minus 1 equals 0. We get natural log of A. Equals to one. And this gives us A equals E. When A equals E, if we check our function again, We would get E to the X divided by 2 E X. Or a limit being 1/2. So F of X. Grows about the same. As G of X, when A is equals C. Now we're considering the values of A that are greater than 0. Now, if you look at this again, We had F of X. As equals a rates to the X, G of X equals to E raise to the X. A race to the X. would grow at a slower rate than eat the excess. This is because A is less than E. We can then tell then That this grows Much slower beneath the X. And we can confirm then that A will be less than E. But it also should be larger than 0. This means that. This will be when 0 is less than a. Which is less than e. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

Exponential growth rates
a. For what values of b > 0 does bˣ grow faster than eˣ as x→∞?

Key Concepts

Exponential Functions

The Number e

Growth Rate Comparison

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