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Which of the following functions grow faster than…

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Problem 2 Medium Difficulty

Which of the following functions grow faster than $e^{x}$ as $x \rightarrow \infty ?$ Which grow at the same rate as $e^{x} ?$ Which grow slower?
a. $10 x^{4}+30 x+1 \quad$ b. $x \ln x-x$
c. $\sqrt{1+x^{4}}$ d. $(5 / 2)^{x}$
e. $e^{-x} \quad$ f. $x e^{x}$
g. $e^{\cos x}$ h. $e^{x-1}$

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Calculus 1 / AB

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Thomas Calculus

Chapter 7

Transcendental Functions

Section 8

Relative Rates of Growth

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Video Transcript

In this example, we have a list of eight different functions that are provided, and our goal is to compare their rate of growth to the function. F of X equals e to the power of X. And there's really three situations. Will have a set of functions that grow slower, some functions that will grow faster and some functions that will have the same rate of growth. So it's begin classifying each one of these functions. The major key is our function. F of X is an exponential function with base E, which is approximately 2.71 So with the first function, we have a degree for Paul. No meal. But polynomial functions always grow slower than an exponential. So we can say why equals 10 to the power of 10 times except power for plus 30 X plus. One must go in the slows grower grow slower category now for the second function. X times natural log X minus X. This is not a polynomial, but the natural log of X grows very, very slow. So this is the most dominant term here, which is just X power one. So for the same reason as before, y equals X natural log X minus X will grow slower than our exponential function, either the ex. Now, when we go to the third function, which is under a square root, we have that. Its most dominant term is something that acts effectively like X squared, considering the square root. And so once again, this is much like a polynomial, and it will grow slower than the exponential. So one plus X to the power four square rooted goes in the grows. Slower column. Let's put It s there now for the next function notice. We're dealing with the exponential this time. So the exponential has base of five halves, which is 2.5, but that 2.5 is less than 2.71 So even though that this is an exponential function, why equals five halves to the par? Vex will also grow slower Onley because its base is smaller. Now let's go to the next function. Each of the power of negative X. This one's tricky because the base is 2.71 Let's consider its graph here. It's a reflection about the Y axis, so its graph would look about like this once we reflect. And as extends tends to infinity, we have that the limit of each of the negative X is zero, which we can see by this graph. And that means that this fifth function y equals the power of negative X will also grow slower. Now we consider why equals X times either the ex. This one's a little bit tricky to compare with either the ex itself. So this time let's take a limit. We're going to consider the limit as X goes to infinity off either the power of negative X divide by x times e to the power of X and hope for the numerator were using u to the power positive X. Rather, when we look at this limit, we can make a cancellation happen here to here. So it's a limit as X goes to infinity of one over X, which is a limit of zero. Now we put either the X in the numerator there, so that tells us that this function ex either the ex here well, in fact, grow faster because we got a limit of zero where it was in the denominator. So the function why equals X times e to the power of X will grow faster than are provided function F of X here. Now let's go to the very next function. With what we've done in mind, we might be hopeful with either the power of co sign X, but co sign X is bounded between negative one and positive one. So this is another example of a function that grows slower. So let's drop it down to this list. You the power of CO sign X for the last situation. We have the same base e, but we have an X minus one any constant next to the X. Whether we're adding, subtracting or even multiplying and dividing does not change the rate of growth. And so we have the function y equals, you know the power of X minus one has the same rate of growth as F of X then that completes the analysis of are given functions

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George B. Thomas, Jr. Maurice D. Weir, Joel Hass

Thomas Calculus

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