23. What roles do the functions e^x and ln(x) play in growth c...

Question

23. What roles do the functions e^x and ln(x) play in growth comparisons?

Accepted Answer

Hello there. Today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Let n be greater than 0, be fixed and consider the functions f of x is equal to the natural log of x. G x is equal to x to the power of n, and h x is equal to e to the power of x, for x is greater than 0. Determine the order of growth of these functions as x approaches infinity. OK, so it appears for this particular problem, we are asked to take all the information that is provided to us and we're asked to determine what the order of growth is for these specific functions, which are specifically the functions of F of X, G X, and H X, and we're asked to determine what the order of growth is of these specific functions as x approaches infinity. So now that we know what we're ultimately trying to solve for, our first step that we need to take is we need to consider the limit as x approaches infinity of the natural log of x divided by x to the power of n. Fantastic. So with that in mind, we need to note that this is an indeterminate form. So when we evaluate it right off the bat, so if we were to plug in a value of infinity in for x, we will know right away that this will yield us an indeterminate form as we will get infinity divided by infinity. As an answer, and because we get an indeterminate form, we need to recall and apply Lahapatal's rule. So when we do that, we will be able to write that the limit as x approaches infinity of the natural log of x divided by x to the power of n. Will be equal to the limit as x approaches infinity of the natural log of x divided by parentheses of x to the power of n, which will be equal to the limit as x approaches infinity of 1 divided by x divided by n multiplied by x to the power of n minus 1. Which will be equal to the limit as x approaches infinity of 1 divided by n multiplied by x to the power of n, which will be simply equal to 0. Our second step now is we need to consider the limit as X approaches infinity of X to the power of N divided by E to the power of X. And in order to consider this limit, we will need to take the logarithms of the ratio, so we will need to take the natural log of x to the n divided by e to the power of x, which will be equal to n multiplied by the natural log of x minus x. And we need to divide by x to get that n multiplied by the natural log of x minus x divided by X will be equal to n multiplied by the natural log of x divided by. X + or -1, it's gonna be -1. Fantastic. So once again, when we divide by X, we will get N multiplied by the natural log of X divided by X minus 1 as a result. And we need to note that since the limit as x approaches infinity of the natural log of x divided by x is equal to 0, that will mean that when we apply Lahapatal's rule, that will mean that the limit as x approaches infinity of 1 divided by x divided by 1 will be also equal to 0. Which will mean, as a result, the right-hand side tends to -1. So let's know that the right-hand side RHS will tend to -1. Therefore, N multiplied by the natural log of X minus X will approach negative infinity as a result. And when we exponentiate, we will find that it will give us a result of x to the power of n divided by e to the power of x will be equal to e to the power of n multiplied by the natural log of x minus x, which will approach 0. Which will mean for our third and final step we can make the following conclusions. From steps 1 and 2 we will obtain that the limit as x approaches infinity of the natural log of x divided by x to the power of n will be equal to 0 and the limit. As X approaches infinity of. X to the power of n divided by e to the power of x will be equal to 0, so that will mean as a result, the growth order as x approaches infinity will be that the natural log of x will be much less than x to the power of n, which will be much less than e to the power of x. And this will be our correct and final answer. Hooray, we did it. So going back to the top to recap what we've learned, we could safely say that our final answer without a doubt will be that once again the natural log of x will be much less than x to the power of n, which will be much less than e to the power of x. And that's it, we've officially solved for this problem. Thank you so much for watching, hopefully that helped, and I can't wait to see you in the next video. bye.

23. What roles do the functions e^x and ln(x) play in growth comparisons?

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