When taking the limit at infinity, is this allowed?

In summary, the video discusses Laplace transforms and mentions the equation \lim_{t \to \infty}\frac{te^{-st}}{-s} being equal to 0. The reasoning for this is that e^t grows faster than t. However, the concept of treating infinity in this way is debated. The conversation also brings up the question of whether \lim_{x \to \infty}\frac{x}{x^2}=0 is valid, to which the response is that \frac{\infty}{\infty} is considered indeterminate. To address this concern, L'Hôpital's Rule can be applied.
  • #1
find_the_fun
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0
In this video of Laplace transforms the equation \(\displaystyle \lim_{t \to \infty}\frac{te^{-st}}{-s}\) is said to be 0. I'm not sure I agree with the reasoning. It says it's because \(\displaystyle e^t\) grows faster than \(\displaystyle t\); can you treat infinity like that? For example could you say \(\displaystyle \lim_{x \to \infty}\frac{x}{x^2}=0\)? I thought it was undefined because obviously \(\displaystyle \frac{infty}{infty}\) is undefined.
 
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  • #2
The form \(\displaystyle \frac{\infty}{\infty}\) is indeterminate...to allay your concerns, apply L'Hôpital's Rule. :D
 
  • #3
MarkFL said:
The form \(\displaystyle \frac{\infty}{\infty}\) is indeterminate...to allay your concerns, apply L'Hôpital's Rule. :D

Maybe the video was aluding to L'Hôpital's Rule when it said \(\displaystyle e^t\) grows faster than \(\displaystyle t\).

By the way, according to here \(\displaystyle \frac{\infty}{\infty}\) is indeterminate.
 

FAQ: When taking the limit at infinity, is this allowed?

What does it mean to take the limit at infinity?

When taking the limit at infinity, we are looking at the behavior of a function as the input values approach infinity, or become infinitely large.

Is taking the limit at infinity always allowed?

No, taking the limit at infinity is only allowed for certain types of functions. Typically, these are functions that approach a constant value, approach positive or negative infinity, or have a horizontal asymptote as x approaches infinity.

How do I know when I can take the limit at infinity?

You can take the limit at infinity when the function satisfies one of the above criteria. Additionally, you can use mathematical techniques such as L'Hopital's rule to determine if a limit at infinity exists.

Does taking the limit at infinity always result in a finite value?

No, taking the limit at infinity can result in different outcomes depending on the behavior of the function. For example, if the function approaches positive or negative infinity as x approaches infinity, the limit will also be infinity. If the function has a horizontal asymptote, the limit will approach that value.

Can taking the limit at infinity be used to find the value of a function at infinity?

No, taking the limit at infinity does not necessarily give us the value of the function at infinity. It only tells us about the behavior of the function as the input values approach infinity. To find the value of a function at infinity, we would need to evaluate the function at a very large number, but this does not necessarily represent the true value at infinity.

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