What Happens to the Limit of This Function as T Approaches Infinity?

In summary: The expression for $E$ still has a $T$ in it when $T\to\infty$.It means that we are looking for an oblique asymptote instead of a horizontal asymptote.That is, if $T\to\infty$ we should find that $\pd E T\to Nk_B$, so that that we get the oblique asymptote $E\to Nk_BT$.
  • #1
Another1
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\(\displaystyle \lim_{{T}\to{\infty}}N \bar{h}\omega \left( \frac{1}{2} + \frac{1}{e^{\frac{ \bar{h}\omega}{k_BT}}-1} \right)\)
In term \(\displaystyle \lim_{{T}\to{\infty}}N \bar{h}\omega \left( \frac{1}{e^{\frac{ \bar{h}\omega}{k_BT}}-1} \right)=N \bar{h}\omega \lim_{{T}\to{\infty}}\left( \frac{1}{e^{\frac{ \bar{h}\omega}{k_BT}}-1} \right)\)
but \(\displaystyle \left( \frac{1}{e^{\frac{ \bar{h}\omega}{k_BT}}-1} \right)\) in the form \(\displaystyle \frac{1}{0}\)
please give me a idea

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  • #2
Hi Another,

The expression for $E$ still has a $T$ in it when $T\to\infty$.
It means that we are looking for an oblique asymptote instead of a horizontal asymptote.
That is, if $T\to\infty$ we should find that $\pd E T\to Nk_B$, so that that we get the oblique asymptote $E\to Nk_BT$.
 
  • #3
Klaas van Aarsen said:
Hi Another,

The expression for $E$ still has a $T$ in it when $T\to\infty$.
It means that we are looking for an oblique asymptote instead of a horizontal asymptote.
That is, if $T\to\infty$ we should find that $\pd E T\to Nk_B$, so that that we get the oblique asymptote $E\to Nk_BT$.

I do not understand, please give an example to explain it?

why $\pd E T\to Nk_B$, ?
 
  • #4
Another said:
I do not understand, please give an example to explain it?

why $\pd E T\to Nk_B$, ?

Consider the function given by $f(x)=\frac 1x + x + 1$.
It has:
$$\lim_{x\to\infty} f(x) = \infty $$

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It also has the oblique asymptote $y=x+1$.
We can find its slope through the limit of $f'(x)$ when $x\to\infty$.
We have:
$$\lim_{x\to\infty} f'(x) =\lim_{x\to\infty} \left(-\frac{1}{x^2} + 1\right) = 1$$
Therefore the oblique asymptote exists and has slope $1$.
Put otherwise, when $x\to\infty$ we have that $f(x)\to x$.
Note that we didn't find the $y$-intercept yet, which we didn't need.
 

FAQ: What Happens to the Limit of This Function as T Approaches Infinity?

What is a limit to infinity?

A limit to infinity is a mathematical concept that describes the behavior of a function as the input approaches an infinitely large value. It is used to determine the value that a function approaches as the input gets closer and closer to infinity.

How do you find the limit to infinity of a function?

To find the limit to infinity of a function, you can use the following steps:
1. Determine the behavior of the function as the input approaches infinity.
2. If the function approaches a specific value, then that is the limit to infinity.
3. If the function approaches infinity or negative infinity, then the limit to infinity does not exist.

What is the difference between a finite limit and a limit to infinity?

A finite limit is the value that a function approaches as the input approaches a specific finite value. A limit to infinity is the value that a function approaches as the input gets closer and closer to infinity.

Can a function have multiple limits to infinity?

No, a function can only have one limit to infinity. If a function approaches different values as the input approaches different infinities, then the limit to infinity does not exist.

How are limits to infinity used in real-world applications?

Limits to infinity are used in various fields such as physics, engineering, and economics to model and predict the behavior of systems as they approach extreme values. They can also be used to analyze the long-term trends and patterns of data.

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