# Understanding Limit: N=1 +/- sqrt(Ae^(2rt))/sqrt(1-Ae^(2rt))

• binbagsss
In summary, the limit where r>0 and A<0 is approached by first writing the expression as N=1±√Ae2rt√1−Ae2rt and then splitting it into two cases depending on the magnitude of A. If |A|<1, the limit is indeterminate and can be simplified to 1±0/1, resulting in a limit of 1. However, if |A|≥1, the limit approaches -∞, as seen when simplifying to 1±√-∞/√1+∞. Despite the numerator not being real, it is no less real than the original form and does not affect the result.
binbagsss

## Homework Statement

[/B]
Trying to understand this limit:

where ##r>0##

## Homework Equations

[/B]
I think it's best to proceed by writing this as:

## N=1 \pm \frac{\sqrt{Ae^{2rt}}}{\sqrt{1-Ae^{2rt}}} ##

## The Attempt at a Solution

[/B]
since ##r>0 ## the exponential term ##\to ## ##\infty## and then since ##A<0## I get two results for ## lim_{t \to \infty} Ae^{2rt} ## depending on ## |A| ##.

a) If ##|A| < 1 ## it goes to zero.
if b) ## |A| \geq 1 ## it goes to ##-\infty##

and where the magnitude of A is not specified in the question.

If it was however for case a) the limit is of an determinate form: ##1 \pm \frac{0}{1} = 1 ##

however for b) i get ## 1 \pm \frac{\sqrt{-\infty}}{\sqrt{1+\infty}}## , and I can't see L'Hopitals rule being much use here due to the square root and exponential terms.

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binbagsss said:

## Homework Statement

[/B]
Trying to understand this limit:
View attachment 217254
where ##r>0##

## Homework Equations

[/B]
I think it's best to proceed by writing this as:

## N=1 \pm \frac{\sqrt{Ae^{2rt}}}{\sqrt{1-Ae^{2rt}}} ##

## The Attempt at a Solution

[/B]
since ##r>0 ## the exponential term ##\to ## ##\infty## and then since ##A<0## I get two results for ## lim_{t \to \infty} Ae^{2rt} ## depending on ## |A| ##.

a) If ##|A| < 1 ## it goes to zero.
if b) ## |A| \geq 1 ## it goes to ##-\infty##

and where the magnitude of A is not specified in the question.

If it was however for case a) the limit is of an determinate form: ##1 \pm \frac{0}{1} = 1 ##

however for b) i get ## 1 \pm \frac{\sqrt{-\infty}}{\sqrt{1+\infty}}## , and I can't see L'Hopitals rule being much use here due to the square root and exponential terms.

$$f(t) = \frac{\sqrt{A e^{2rt}}}{\sqrt{1-Ae^{2rt}}} = \frac{e^{-rt}}{e^{-rt}} \frac{\sqrt{A e^{2rt}}}{\sqrt{1-Ae^{2rt}}} = \frac{\sqrt{A}}{\sqrt{e^{-2rt} - A} } \to \frac{\sqrt{A}}{\sqrt{-A}}$$

Ray Vickson said:
$$f(t) = \frac{\sqrt{A e^{2rt}}}{\sqrt{1-Ae^{2rt}}} = \frac{e^{-rt}}{e^{-rt}} \frac{\sqrt{A e^{2rt}}}{\sqrt{1-Ae^{2rt}}} = \frac{\sqrt{A}}{\sqrt{e^{-2rt} - A} } \to \frac{\sqrt{A}}{\sqrt{-A}}$$
but A<0, so the numerator is not real?

binbagsss said:
but A<0, so the numerator is not real?

Right, but no less real than the original form ##\sqrt{A e^{2rt}}##.

Last edited:
Ray Vickson said:
Right, but no less real than the original form ##\sqrt{A e^{2rt}}##.

so then do we not get ## 1 \pm i ## rather than ## 1 \pm 1 ## as in the solution above?

## What is "Quick Limits Question"?

"Quick Limits Question" is a term used to describe a type of question that is focused on determining the maximum or minimum value of a function or equation. These types of questions often involve finding the limit of a function as a variable approaches a specific value.

## Why are "Quick Limits Question" important?

Understanding how to solve "Quick Limits Question" is important in many fields of science, including mathematics, physics, and engineering. These types of questions allow scientists to determine the behavior of a function near a specific point, which can be useful in making predictions and solving real-world problems.

## What are some common techniques for solving "Quick Limits Question"?

Some common techniques for solving "Quick Limits Question" include using algebraic manipulation, using L'Hospital's rule, and graphing the function. Other methods such as substitution and using the limit laws can also be helpful in finding the limit of a function.

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One common mistake when solving "Quick Limits Question" is forgetting to check for any discontinuities or asymptotes in the function. It is important to analyze the function and its behavior near the given point before attempting to solve the limit. Additionally, not properly simplifying the function or incorrectly using limit laws can also lead to errors.

## Can "Quick Limits Question" be applied to real-world situations?

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