# Unfamiliar algebra within L'Hopital

• LearninDaMath
In summary, when simplifying an expression involving a limit, it is helpful to look for ways to turn it into a rational expression and then simplify the numerator if necessary. In this specific conversation, factoring out e^(2x) from the denominator was done to avoid using the chain rule when applying L'Hopital's rule. This can also be applied to future problems to make them easier to work with.
LearninDaMath

## Homework Statement

$\stackrel{lim}{x\rightarrow}0^{+}$ ($e^{2x}-1)^{x}$

At the step of dividing by e^x, how does that algebra work?

e^(2x) was factored out from the denominator. In the step right before the blue box, we can factor out e^(2x) from (e^(2x) - 1) so we have [e^(2x)](1 - (1/e^(2x))) = [e^(2x)](1 - e^(-2x)):

$\frac{e^{2x}}{e^{2x} - 1} = \frac{e^{2x}}{e^{2x}(1 - \frac{1}{e^{2x}})} = \frac{1}{1 - e^{-2x}}$.

Thanks stripes, appreciate the illustration with the latex reference as well.

I suppose that the reason for factoring out and cancelling away e^2x from this rational function is to avoid having to do an unnecessary product rule (and or chain rule) when applying L'hopitals for the second time. Would this be right?

For future problems similar to this one, what should I be thinking when doing these problems so as to notice when something can be factored out in order to make the next steps easier? Should I just look for similar terms in the numerator and denominator, and if there are, just go ahead and cancel them away before proceeding to any next steps?

Last edited:
Yes, doing this eliminates the need for the chain rule in the numerator, so by cancelling the e^(2x), we are simplifying the numerator, and completely eliminating the denominator (i.e., turning the denominator to 1), and thereby making the entire expressing non-indeterminate form.

The best thing to do is to try to turn the expression into a rational one (a ratio of two expressions), and then trying to simplify the numerator if necessary. Sometimes you will be able to take the derivative of the top and bottom easily, but when you cannot, then try to manipulate the expression a little bit in your favor. Look for similar terms, anything that can lead to an expression that is easier to work with.

Unfortunately, depending on the problem, this might be easier said than done. But perseverance is key!

## 1. What is unfamiliar algebra within L'Hopital?

Unfamiliar algebra within L'Hopital refers to the use of algebraic manipulations to simplify and solve indeterminate forms in calculus problems, particularly when using L'Hopital's rule. It involves using algebraic properties and techniques to transform an indeterminate form into a form that can be evaluated easily.

## 2. How is unfamiliar algebra used in L'Hopital's rule?

L'Hopital's rule is a calculus technique used to evaluate limits involving indeterminate forms such as 0/0 or infinity/infinity. Unfamiliar algebra is used to manipulate the given function in such a way that the indeterminate form can be transformed into a simpler form that can be evaluated using L'Hopital's rule.

## 3. What are some common algebraic techniques used in L'Hopital's rule?

Some common algebraic techniques used in L'Hopital's rule include factoring, rationalizing the denominator, using the properties of logarithms and exponents, and simplifying fractions by canceling common factors. These techniques help to transform an indeterminate form into a form that can be evaluated easily.

## 4. Why is it important to understand unfamiliar algebra within L'Hopital?

Understanding unfamiliar algebra within L'Hopital is important because it allows for the evaluation of indeterminate forms in calculus problems and can help to simplify complex expressions. It also helps to develop a deeper understanding of algebraic concepts and their applications in calculus.

## 5. How can I improve my skills in unfamiliar algebra within L'Hopital?

To improve your skills in unfamiliar algebra within L'Hopital, it is important to practice solving problems involving indeterminate forms, using algebraic techniques to simplify expressions. You can also review algebraic concepts and properties, as well as familiarize yourself with common indeterminate forms and their solutions using L'Hopital's rule.

Replies
1
Views
750
Replies
14
Views
1K
Replies
2
Views
1K
Replies
13
Views
2K
Replies
6
Views
744
Replies
21
Views
2K
Replies
3
Views
2K
Replies
5
Views
1K
Replies
4
Views
1K
Replies
4
Views
1K