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?

a. 10x^4 + 30x + 1

Accepted Answer

Welcome back everyone. Determine whether the function F of X equals 4 X cubed minus 10 X + 1 grows faster than E to the power of X as X approaches infinity. A says yes and B says no. For this problem, let's assume. That the answer is yes, so that we have a reference point. If F of X grows faster than eats the power of X, then we can conclude that the limit is X approaches infinity. Of e to the power of X divided by F of X is equal to 0. Why is this the case? Well, it basically represents that denominator dominates over the numerator so that the limit is 0, and therefore, the denominator function grows faster. Let's check if this limit is indeed 0. What we're going to do is substitute F of X. So we're going to get the limit as X approaches infinity of E to the power of X, divided by 4 X cubed minus 10 X plus 1. So in this case, if we simply substitute infinity, we're going to get infinity. In the numerator, and then 4 multiplied by infinity cubed minus 10 infinity plus 1, this gives us infinity in the denominator. Right? Because X cubed dominates over X, so infinity cubed minus infinity basically results in positive infinity. This indeterminant form means that we can apply Lapito's rule. So let's go ahead and differentiate the numerator and the denominator. We're going to get E to the power of X in the numerator, because the derivative of E to the power of X is equal to itself. And for the denominator, the derivative of X cubed is 3 X2, so we're going to get 12 X2 minus. Because the derivative of X is equal to 1 and the derivative of 1 is 0. So once again, if we substitute X approaches infinity, we're going to get infinity in the numerator. And in the denominator, that's 12 multiplied by infinity squared minus 10, which is infinity. Since we still have that form, we can apply Lapito's rule once again. We're going to get the limit as X approaches infinity. The derivative of the power of X is the power of X, and in the denominator, we want to differentiate X2, which is 2 X. 2 X multiplied by 12 gives us 24 X. The derivative of -10 is 0, it's a constant. Once again we substitute X approaches infinity to get infinity divided by 24 multiplied by infinity, which is infinity. So once again we apply a laps rule. To get the limit as x approaches infinity of E to the power of X divided by 24. And in this case, we can clearly see that we are dividing infinity by 24, which is basically infinity, right? So that we did not get 0, meaning the answer is no. If we get infinity, it basically means that the numerator function grows faster than the denominator. So in fact, eats the power of X grows faster than F of X, and therefore, the answer to this problem is B. No. F of X does not grow faster than it eats the power of X as X approaches infinity. 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?a. 10x^4 + 30x + 1

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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?
a. 10x^4 + 30x + 1