10. True, or false? As x→∞,
a. 1/(x+3) = O(1/x)

Question

10. True, or false? As x→∞,

a. 1/(x+3) = O(1/x)

Accepted Answer

Hello. In this video we are going to be determining whether the given statement is true or false. We are told that x + 1 divided by x2 + 4 is equal to the rate of growth of 1 divided by x. In order to answer this, let's go ahead and determine or review the definition of the rate of growth. The definition says the following if f of x. is equal to the rate of growth of another function g of x. As x approaches infinity. There exists an integer m that's greater than 0 and a starting value x not such that For all values of x. That is greater than or equal to the starting value x. The absolute value of F of X is less than or equal to m multiplied by the absolute value of G of X. So, what we want to go ahead and do is we want to go ahead and pick an Xal value to start with. Let's go ahead and start with X is equal to 1. When x is equal to 1 and taking a look at the denominator of the function, we get that x2 + 4 must be greater than or equal to x2. So our function x + 1 divided by x2 + 4 must be greater than or equal to 0, but it must also be less than or equal to x + 1 divided by x2, and x + 1 divided by x2 is equivalent to 1 divided by x plus 1 divided by x2. So by the definition we know that when x is greater than equal to 1. Then 1 divided by x 2. Must be less than or equal to 1 divided by x. Therefore, 1 divided by x plus 1 divided by x 2 must be less than or equal to 1 divided by x plus 1 divided by x, which is less than or which is equal. To 2 to 2 divided by X. And so what this means is that our function, absolute value of x + 1 divided by x2 + 1 must be less than or equal to 2 multiplied by the absolute value of 1 divided by x, and this is where m is equal to the value of 2. So because this is true for when m is equal to 2, that means that the statement is going to be true. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

10. True, or false? As x→∞,a. 1/(x+3) = O(1/x)

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10. True, or false? As x→∞,
a. 1/(x+3) = O(1/x)