If a function f represents a system that varies in time, the e...

Question

If a function f represents a system that varies in time, the existence of lim $\lim_{t \to \infty} f (t)$ means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady state exists and give the steady-state value.

The population of a bacteria culture is given by $p (t) = \frac{2500}{t + 1}$ .

The population of a bacteria culture is given by $p (t) = \frac{2500}{t + 1}$ .

Accepted Answer

Welcome back everyone. The concentration of a chemical in a reaction over time is modeled by the function C of T equals 500 divided by two T plus five, determine whether a steady state exists and provide its value. As T approaches infinity. A says the steady state exists and its value is zero. B says it exists and its value is 250 C, it exists and its value is 500. And D says the steady state does not exist. Now to determine if a steady state concentration exists. As T approaches infinity, we can evaluate the limit of the function C of T as T approaches infinity. In other words, OK, we can find the limit as T approaches infinity of C of T which is going to be equal to the limit. OK? Of 500 divided by two T plus five. Not to simplify our expression, we can divide all the terms in the numerate and numerator and the denominator by T, the highest power of T in the denominator. So no, we can rewrite this as the limit. As T approaches infinity of 500 divided by T divided by two T divided by T plus five divided by T, we can simplify this expression. OK. For our T is to cancel to have 500 divided by T divided by two plus five divided by T. Now, why is this helpful? Well, as T approaches infinity, we can evaluate the limits for our terms, 500 divided by T and five divided by T because in both of these terms, as T gets arbitrarily large, our quotient or fraction is going to tend to zero. OK. So in other words, we can rewrite this then or the limit then is going to be equal to zero, divided by two plus 02 plus zero equals two and zero, divided by two equals zero. Th as T approaches infinity, the concentration C of T approaches zero. Therefore, a steady state does exist and its value is zero. If we look back at our answer choices, that means A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Limits

Steady State

Population Dynamics

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