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?

c. √x

Accepted Answer

Welcome back, everyone. Decide whether the function F of X equals eats the power of 1/2 x grows faster than eats the power of X as X approaches infinity. He says eats the power of 1/2 x grows slower than eats the power of X. eats the power of 1/2 x grows faster than eats the power of X. C eats the power of 1/2 x and eats the power of x grow at the same rate and the growth comparison is inconclusive. What we're going to do in this problem is identify the limit of the ratio of the two functions. We're going to take a limit as X approaches infinity, and now it doesn't matter if we take F divided by G or G divided by F. We just have to be rational, right? So, let's go ahead and take F of X divided by G of X. Now if G grows faster, then our limit must be equal to 0, right? If F grows faster, we're going to get a limit of infinity, and if we get a constant, well, they grow at the same rate. So let's go ahead and substitute the functions. We get limit as X approaches infinity. F of X is eat to the power of 1/2 X. And G of X is. E to the power of X. If we simply substitute infinity for each X, we're going to get infinity divided by infinity, which is an indeterminate form. So let's simplify using the properties of exponouns. We can rewrite it as limit as x approaches infinity of E to the power of 1/2 x minus X. We're subtracting the two exponents, and this reduces to the limit as X approaches infinity. Of e to the power of negative 1/2 X. we can also write it as limit. As x approaches infinity of 1 divided by e to the power of positive 1/2 x, and now we are dividing 1 by infinity because our exponent approaches infinity and it's the power of infinity is infinity. So a constant divided by an infinitely large number is going to be 0. That's our limit, right? Because the limit is 0, we have shown that G of X dominates against F of X, and therefore G of X grows faster. Meaning it to the power of 1/2 X. Girls Slower Then our denominator, which is G of X, right? So slower down. E to the power of X and therefore the correct answer to this problem corresponds to the answer choice a. Let's label it. 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?c. √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?
c. √x