3. Which of the following functions grow faster than x² as x→∞...

Question

3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?

g. x^3 e^(-x)

Accepted Answer

Welcome back everyone. As x approaches infinity, which function, if any, grows faster? F of x equals x to the power of 4 eats the power of neg x divided by 2, where g of x equals x 2. A says f of x, bg of x. C, they grow at the same rate, and d cannot be determined. For this problem, we're comparing the growth rate and we want to identify the limit as x approaches infinity of the ratio of the two functions. So we're going to take F of x divided by g of x. Substituting, we're going to write limit as x approaches infinity of x the power of 4, e to the power of negative x divided by 2, divided by x 2. And immediately we can divide x the power of 4 by x 2 to get the final function x 2 e to the power of x divided by 2 within our limit. Using the properties of exponents, we can rewrite it as. Limit as x approaches infinity of x 2 divided by e to the power of x divided by 2 because we have a negative exponent, so we can put e. Raise the power of x divided by 2 into the denominator. And now as x approaches infinity, the numerator is x 2, which is going to be infinitely large, and e to the power of infinity divided by 2 is going to approach infinity as well. So we get an indeterminate form infinity divided by infinity, meaning we can apply Lapel's rule, which says that we want to differentiate the numerator and the denominator. The derivative of the numerator is 2x using the power rule. In the denominator we're going to apply the chain rule to get e to the power of x divided by 2 multiplied by the derivative of 1/2 x, right? Since the derivative of x is 1, we simply get 1/2. And now let's go ahead and simplify. This is equal to limit as x approaches infinity. 2 divided by 1/2 is equal to 4, so we get 4 x divided by e to the power of x divided by 2. And we will still get infinity divided by infinity because we have X divided by e to the power of X divided by 2.4 simply has no effect because we can factor it out, right? So, once again, let's apply Lapito's rule. By differentiating once more. In the numerator, the derivative of x is 1, so we simply get 4, and in the denominator we already know that the derivative is 1/2 eats the power of x divided by 2. Performing division, we can write it as limit as x approaches infinity of 4 divided by 1/2 is 8. And 8 is divided by e to the power of x divided by 2. And now this is 8 divided by an infinitely large number. This limit is equal to 0 because we have a constant divided by an infinitely large number, right? Whenever we get a limit of zero, it basically means that the denominator function dominates over the numerator function and therefore g of x in this case grows faster, so the correct answer is option B. Thank you for watching.

3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?g. x^3 e^(-x)

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3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
g. x^3 e^(-x)