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.

a. f(x) = 3^(-x), g(x) = 2^(-x)

Accepted Answer

Welcome back everyone. As x approaches infinity, which function decays faster? F of x equals 5 of x, or g x equals 2 x. For this problem, let's remember that whenever we are comparing the growth rate of two functions, we basically want to find the limit as x approaches infinity of their ratio. Notice that in this problem, we're looking for the decay, right? So, it's opposite to growth rate. Meaning, first of all, we, we will identify the growth and then we will conclude something that is opposite. So, let's begin by identifying the limit as x approaches infinity of the ratio of two of the two functions. We can take F in the numerator and G in the denominator. Substituting the functions, we're going to get the limit as x approaches infinity of 5 to the power of x divided by 2 to the power of x. To identify this limit, we can use the properties of exponents. We can basically show that this is limit as x approaches infinity of 5 halves raised to the power of negative x. And now we can turn our exponent into positive x by taking the reciprocal of 5 halves, so we're going to get the limit as x approaches infinity of 2/5 to the power of x. Now, notice that X approaches infinity and we have a number between -1 and 1, right, which is 2/5. So when X approaches infinity, And if we are raising a number that is between -1 and 1 to an infinitely large exponent, the limit is going to be equal to 0. Remember that limit as x approaches infinity of. Some number a to the power of x is equal to 0 as long as the absolute value of a is less than 1. So this is the case for this limit we have shown that this limit is equal to 0. And remember, the limit is equal to 0 whenever g of x dominates over f of x in terms of growth rates, meaning g of x grows. Faster Or basically we can say that F of X decays faster. Right? Because we're comparing the growth rates, and if we want to understand decay, it's basically the opposite to growth. So we can conclude that F of X decays faster than G. This 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.a. f(x) = 3^(-x), g(x) = 2^(-x)

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