110. Does f grow faster, slower, or at the same rate as g as x...

Question

110. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.

c. f(x) = 10x^3 + 2x^2, g(x) = e^x

Accepted Answer

Welcome back, everyone. Determine which function grows faster as X approaches infinity. F of X equals 3 X to the power of 5 + 7 X, or G of X equals e to the power of X. So for this problem we want to identify the limit as x approaches infinity of the ratio of two functions. I suppose that we take F for the numerator and G for the denominator. So we are going to get the limit as X approaches infinity of 3 X to the power of 5 + 7 X divided by e to the power of X. And now if we substitute infinity, we're going to get infinity divided by infinity. So what we can do is simply applied the Lapito's rule, right? Let's remember that we're differentiating the numerator and the denominator, and specifically whenever we differentiate our denominator, it is always going to be E to the power of X. While our numerator is going to remain a polynomial. Specifically, we're going to differentiate X to the power of 5, which is X the power of 4, and our numerator is still going to approach infinity. So what we want to do in the numerator is simply differentiated 5 times, right? Let's remember the rule. First of all, because we have a constant which is 3, we can factor it out and differentiating X to the power of 5. 5 times, we're going to get 5 factorial, right, while 7 X is going to vanish because differentiating it once we are going to get 7, and the derivative of 7 is basically 0, or our denominator, we're going to have E to the power of X for any derivative. So now we have a constant divided by an infinitely large number which is equal to 0. What does that mean? Well, essentially, because this limit is 0, it essentially means that the denominator function grows faster than the numerator function. So we can conclude that she grows. Faster? Then which is our final answer for this problem. Thank you for watching.

110. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.c. f(x) = 10x^3 + 2x^2, g(x) = e^x

Watch next

110. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
c. f(x) = 10x^3 + 2x^2, g(x) = e^x