2. Which of the following functions grow faster than e^x as x→...

Question

2. 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. e^(-x)

Accepted Answer

Welcome back everyone. Which function grows faster as X approaches infinity? E to the power of 2 X or eat to the power of X? A says E to the power of 2 X. B eats the power of X. C. They grow at the same rate, and D cannot be determined. For this problem, let's remember that if we want to compare the growth rates of two functions, we simply want to take their ratio. So we can begin with Es the power of 2 X and divide by Es the power of X, and we're taking the limit of this ratio as X approaches infinity. If we simply substitute infinity for X, we're going to get e to the power of infinity, which is infinity. Divided by its the power of infinity, which is also infinity, right? So what we can do is simply apply the properties of exponents to rewrite this limit as x approaches infinity of E to the power of 2 X minus X, which gives us a limit as x approaches infinity of E to the power of X, and now we do not have an indeterminate form anymore. Because that's e to the power of infinity, which is equal to infinity. So now, if this limit is infinity, it basically means that the numerator function dominates over the denominator function, right? So the numerator function grows faster than the denominator function. If we get 0 on the other hand, it basically means that the denominator function grows faster. If we get a constant that is not equal to 0, then they grow at the same rate. So, based on the fact that we got infinity, it means that the numerator function, which is E to the power of 2 X. Gros Faster Then the denominator function, which is eats the power of X. So the answer to this problem is A eats the power of 2 X grows faster. Thank you for watching.

2. 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. e^(-x)

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2. 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. e^(-x)