# Limit of Newton's Law of Cooling....1

• MHB
In summary, the limit of u(t) as t tends to positive infinity is T, as given by the equation u(t) = (u_0 - T)e^(kt) + T and the condition that k<0. As t approaches infinity, the exponential term e^(kt) goes to 0, resulting in T as the limit of u(t). This can be understood by reading the textbook and studying the concept of limits.
Given u(t) = (u_0 - T)e^(kt) + T, find the limit of u(t) as t tends to positive infinity.

The answer is T. How is the answer found? Seeking a hint or two.

Beer soaked ramblings follow.
Given u(t) = (u_0 - T)e^(kt) + T, find the limit of u(t) as t tends to positive infinity.

The answer is T. How is the answer found? Seeking a hint or two.
Problem 1.5.75.a.
Some details left out.
Suggest you post a screenshot of the entire problem instead of making helpers guess the condition for k.

Given u(t) = (u_0 - T)e^(kt) + T, find the limit of u(t) as t tends to positive infinity.

The answer is T. How is the answer found? Seeking a hint or two.
That is simply NOT true! as t goes to positive infinity, e^{kt} goes positive infinity. u(t) would go to negative infinity, not T. It is true if t tended to negative infinity or if the exponential were e^(kt).

Country Boy said:
That is simply NOT true! as t goes to positive infinity, e^{kt} goes positive infinity. u(t) would go to negative infinity, not T. It is true if t tended to negative infinity or if the exponential were e^(kt).

I posted the answer given in the textbook. Let me look screen shot the question.

Look at 75 parts (a) and (b).

Beer soaked ramblings follow.
Country Boy said:
That is simply NOT true! as t goes to positive infinity, e^{kt} goes positive infinity. u(t) would go to negative infinity, not T. It is true if t tended to negative infinity or if the exponential were e^(kt).
As I intimated, nycmathdad omitted the detail that k<0.

Now that we know my error (or should I call it sin), how is 75 done?

Beer soaked ramblings follow.
Now that we know my error (or should I call it sin), how is 75 done?
If you have done your reading, it should be clear to you what $\lim_{t \to \infty} e^{kt}$ if k<0.
If not, I suggest you whip out your calculator and try out values for increasing values of t for a specific constant value of k<0 like -1/2.

In 75, you are given that $$\displaystyle u(t)= (u_0- T)e^{kt}+ T$$ AND it is specified that k< 0. As t goes to infinity $$\displaystyle e^{kt}$$ goes to 0 so u(t) goes to $$\displaystyle (u_0- T)(0)+ T= T$$

Country Boy said:
In 75, you are given that $$\displaystyle u(t)= (u_0- T)e^{kt}+ T$$ AND it is specified that k< 0. As t goes to infinity $$\displaystyle e^{kt}$$ goes to 0 so u(t) goes to $$\displaystyle (u_0- T)(0)+ T= T$$

I don't understand why e^(kt) goes to 0 as t goes to positive infinity. In other words, why does e^(kt) become zero?

Beer soaked ramblings follow.
I don't understand why e^(kt) goes to 0 as t goes to positive infinity. In other words, why does e^(kt) become zero?
For every action, there is an equal and opposite reaction.
If you didn't read your book, then you you won't understand why.

Thank you everyone. I will look further into this chapter.

## 1. What is Newton's Law of Cooling?

Newton's Law of Cooling is a scientific law that describes the rate at which an object cools down when placed in a different temperature environment. It states that the rate of change of temperature of an object is proportional to the difference in temperature between the object and its surroundings.

## 2. What is the limit of Newton's Law of Cooling?

The limit of Newton's Law of Cooling is when the temperature difference between the object and its surroundings becomes negligible, meaning that the object and its surroundings have reached thermal equilibrium. At this point, the rate of change of temperature becomes zero and the object's temperature remains constant.

## 3. How is Newton's Law of Cooling used in scientific research?

Newton's Law of Cooling is used in scientific research to study the cooling rates of various objects and substances. This information can be used to understand the properties of different materials and to develop more efficient cooling systems.

## 4. Is Newton's Law of Cooling applicable to all objects?

No, Newton's Law of Cooling is only applicable to objects that are in a closed system and have a constant temperature difference with their surroundings. Objects that are in an open system or have varying temperature differences may not follow this law.

## 5. Can Newton's Law of Cooling be used to predict the exact temperature of an object?

No, Newton's Law of Cooling can only provide an approximate prediction of an object's temperature. This is because there are other factors, such as air flow and insulation, that can affect the cooling rate of an object and are not accounted for in this law.

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