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

Question

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

c. x²e^(-x)

Accepted Answer

Welcome back everyone. As x approaches infinity, which function, if any, grows faster? F of x equals x the power of -2 e to the power of x, or g x equals x 2. A f of x, bg of x. C, they grow at the same rate, and d cannot be determined. For this problem, let's remember that to compare the growth weight of two functions we identify the limit as x approaches infinity of their ratio. We can take f of x divided by g of x. Let's substitute each function. Which gives us limit as x approaches infinity of x to the power of -2 e to the power of x divided by x 2. And what we can do is simply use the properties of the exponents to rewrite our expression as follows Limit as x approaches infinity. In the denominator we're going to have x the power of 2 because -2 brings x into the denominator. We can turn our exponent from negative to positive, which gives us x2 multiplied by x2 multiplied by e to the power of x once again using the reciprocal rule. E to the power of negative x is 1 divided by e to the power of positive x, meaning in the numerator we simply have 1, and in the denominator we have x 2 multiplied by x2 multiplied by e to the power of x. And then once again we can use the properties of exponents to rewrite x2 multiplied by x2 as x the power of 4. So now we want to evaluate the limit as x approaches infinity. Of 1 divided by x, the power of y eats the power of x. As x approaches infinity, our enumerator remains 1 because it's a constant. Our denominator is going to be infinity to the power of 4, which is infinity, multiplied by each the power of infinity, which is also infinity. So basically we get a constant divided by infinity. And this means the limit is 0 because whenever we are dividing a constant by an infinitely large number. This yields a element of 0. And whenever we get a limit of 0 for this ratio, it basically means that the denominator function. Dominates over the numerator function and grows faster, so the correct answer for this problem is option B. G of x grows faster. Thank you for watching.

4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?c. x²e^(-x)

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4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
c. x²e^(-x)