{Use of Tech} Population growth Consider the following populat...

Question

{Use of Tech} Population growth Consider the following population functions.

d. Evaluate and interpret lim t→∞ p(t).

p(t) = 600 (t²+3/t²+9)

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Consider the following revenue function of time for a company. Evaluate and interpret the limit evaluated as T approaches infinity of R of T. R of T is equal to 800 multiplied by T 2 + 5 divided by T 2 + 11. Awesome. So it appears for this particular problem, we're asked to evaluate and interpret the limit evaluated at T approaches infinity of RFT given this expression for the function for RFT. So with that said, our first step in order to solve this problem is we need to identify the revenue function, which is RT and RT, our revenue function. So that's our revenue function is R of T, and this is the target varier. So this is the target variable that we're trying to solve for, and this will be the limit, so which is the limit R of T as T approaches infinity, so meaning we could go ahead and safely write that the limit as T approaches infinity. Of 800 multiplied by T like T2 + 5 divided by T2 + 11. Awesome, and don't forget your parenthesis. So now we need to evaluate the limit as T approaches infinity of R T, and we need to divide the numerator and the denominator of the fraction inside the function by T to the power of 2. So when we do that, we will get that the limit as T approaches infinity of. 800 multiplied by T power of 2 + 5 divided by T power of 2. Plus 11. is equal to 800 multiplied by the limit as T approaches infinity of T to the power of 2 + 5 divided by T 2. Divided by T 2 + 11, divided by T power of 2. Fantastic, and we do a little bit of simplifying, we will find that this is equal to 800 multiplied by the limit as T approaches infinity of 1 plus. 5 divided by T 2 divided by 1 + 11 divided by T to the power of 2. And once again, don't forget your parenthesis. Fantastic. So now our next step. we need to evaluate the limit. So we need to note that the fractional terms, 5 divided by T 2 and 11 divided by T to the power of 2, will both approach 0 as T approaches infinity. So let's make a little note of that. So we need to note that 5 so we need to take 5. Divided by T 2 and 11 divided by T 2 will both approach infinity. And they will both approach, sorry, they will both approach 0, so this is important, so they'll both approach 0 as T approaches infinity. So that's important. So these two fractional terms will approach 0 as T approaches infinity. Awesome. So with that said, We could go ahead And write that 800 multiplied by the limit as T approaches infinity of 1 + 5 divided by T 2 divided by 1 + 11 divided by T the power of 2. Is equal to 800. Multiplied by and when we plug in a value of 0 in for T, we will get 1 + 0, or rather, when we plug in infinity in for T, we'll get 1 + 0 divided by 1 + 0. And once again we're plugging in infinity in for the value for T because we're evaluating T as it approaches infinity. And when we do that, we will find that it's equal to 800. Hooray. So, therefore, we can conclude that the limit of the revenue function as T approaches infinity is 800. Which means that the revenue stabilizes at 800 units in the long run. So therefore we could safely write that the limit as T approaches infinity of R of T is equal to 800. And we can also simply state that. The revenue. Will stabilize, so the revenue stabilizes. At 800 units. In the long run, in the long run. And that's it. That is our final answer. Hooray, we did it. So to quickly recap here, Our final answer for the limit is the limit as T approaches infinity of RFT is equal to 800, and the revenue stabilizes at 800 units in the long run. And that's it. That is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

{Use of Tech} Population growth Consider the following population functions.
d. Evaluate and interpret lim t→∞ p(t).
p(t) = 600 (t²+3/t²+9)

Key Concepts

Limits

Population Functions

Asymptotic Behavior

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