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.

e. f(x) = arccsc(x), g(x) = 1/x

Accepted Answer

Welcome back everyone. Determine which function grows faster as x approaches infinity. F of x equals inverse of sin of 3 divided by x. 4 g of x equals 3 divided by x. A says F grows faster than G. B says G grows faster than F. C. They grow at the same rate, and D cannot be determined. Whenever we went to Compare the growth rate of two functions. We basically want to find the limit as x approaches infinity of their ratio. We can say k x divided by g of x. So let's substitute each function. We're going to get the limits x approaches infinity of inverse of sine of 3 divided by x divided by 3 divided by x. Now, to solve for this limit, we can introduce a substitution. Let's suppose that z is equal to 3 divided by x. So now, as x approaches infinity, right, we want to ensure that Z, which is 3 divided by x. is going to approach 0 from the right, right? Because if x approaches positive infinity, we're going to divide 3 by an infinitely small number that is positive, right? So we want to ensure that we're approaching Z, I'm sorry, we're approaching 0 from the right. So in this case our limit becomes limit as z approaches 0 from the right of inverse of sine of z divided by z. And now let's remember that by definition this limit is equal to one. Now that we know that our limit is equal to 1, notice that it is a constant which is not equal to 0. Whenever we get a limit of a constant that is not equal to 0, we can conclude that the two functions grow at the same rate. So the correct answer for this problem corresponds to the answer choice C. Thank you for watching.

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