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?

a. x² + √x

Accepted Answer

Welcome back everyone. As x approaches infinity, which function, if any, grows faster? F of x equals x cubed plus square root of x, or g x equals x cubed. A S F X B G X C, they grow at the same rate and d cannot be determined. For this problem, let's remember that in order to compare the growth rate of two functions. We want to find the limit as x approaches infinity of their ratio. We can take f of x in the numerator and g of x in the denominator. Substituting, we are going to get the limits x approaches infinity of x cubed plus square root of x divided by x cubed. We can also express square root of x as x the power of 1/2, and we can perform division using the properties of exponents. Because we have the same base in each case, we can subtract the exponents. x cubed divided by x cubed gives us 1 plus x 1/2 divided by x cubed gives us x 1/2. -3. Using the common denominator, we get a 1 minus 6, which is -5, divided by 2. So the overall exponent is going to be -5 halves. We can also express that term as 1 divided by x to the power of 5 halves. And this is the limit that we want to find. Now 1 divided by x, the power of 5 halves as x approaches infinity, is going to approach 0 because we have a constant divided by an infinitely large number and therefore the limit is equal to our constant which is 1. Remember that whenever a limit of this ratio is a constant that is not equal to 0, the two functions grow at the same rate. So the correct answer for this problem corresponds to the answer of choice C. 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?a. x² + √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?
a. x² + √x