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

Question

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

b. f(x)=x, g(x)=x + 1/x

Accepted Answer

Welcome back, everyone. Compare the growth rates as X approaches infinity of the functions F of X equals X2 and G of X equals X2 + 3 X. For this problem, let's remember that whenever we want to compare the growth rates of two functions, we want to find the limit as X approaches infinity of their ratio. We're going to take G of X in the numerator and F of X in the denominator, which gives us limit as X approaches infinity of X2 + 3 x divided by X2. If we simply attempt the direct substitution of an infinitely large number, we're going to get infinity divided by infinity, which is an indeterminate form. So to identify the actual value of the limit, we're going to perform division. X squared divided by X2 gives us 1. And 3 x divided by X squared leaves us with 3 divided by X. Now 3 divided by X approaches 0 as x approaches infinity because we have a constant divided by an infinitely large number, and we are left with a limit of a constant, which is that constant itself. So the limit is 1. Let's understand that our limit is a constant. That is not equal to 0. Whenever we get an actual constant that is not equal to 0, we can conclude that the two functions grow at the same rate. So we can say that F. Gross At the same rate. SG, and that's our final answer for this problem. Thank you for watching.

109. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.b. f(x)=x, g(x)=x + 1/x

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109. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
b. f(x)=x, g(x)=x + 1/x