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?

g. (1.1)^x

Accepted Answer

Welcome back everyone. As x approaches infinity, which function, if any, grows faster? F of x equals t to the power of x or g 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 to compare the growth rate of the two functions, we want to identify the limit as x approaches infinity of their ratio. So we're going to take f of x in the numerator and g of x in the denominator. Which gives us a limit as x approaches infinity of 2 to the power of x divided by x 2. If we simply substitute infinity for x, we're going to get t to the power of infinity, which is infinity, divided by infinity squared, which is also infinity, and this is an indeterminate form. For this indeterminate form we can apply Lapel's rule. According to Lapiel's rule, we want to differentiate the numerator. The derivative of t to the power of x is t to the power of xl of 2. And we want to differentiate the denominator as well, the derivative of x 22 x. And once again we are going to attempt direct substitution which gives us infinity multiplied by eon of 2 that's still infinity divided by 2 multiplied by infinity, which is infinity. Since we still have the same indeterminate form, we can apply Lapto's rule once again. LN of 2 is a constant, so we can just differentiate 2 to the power of x, which is 2 to the power of x, L of 2. And since we're multiplying by ln of 2, which is a constant, we can just write 2 to the power of xln 2 of 2. In the denominator, the derivative of x is 1, so we get 2 multiplied by 1, which is simply 2. Now in the numerator, we're going to get infinity multiplied by ln2 of 2. Now ln 2 of 2 is a constant, so we get infinity. Divided by 2. Which is basically infinity. Since we got a limit of infinity, it basically means that the numerator function f x dominates over the denominator function, and f x grows faster. So the correct answer for this problem corresponds to the answer choice a f x grows faster than g of x. 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?g. (1.1)^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?
g. (1.1)^x