# L'hopital's rule, indeterminate forms

• Panphobia
In summary, the conversation discusses finding the limit of the expression (x/(x+1))^x as x approaches infinity. The suggested method involves using l'Hospital's rule and solving for y by taking the natural logarithm of the expression. The speaker suggests trying the process again to find the correct answer.
Panphobia

## Homework Statement

$lim_{x -> \infty} \left( \frac{x}{x+1} \right) ^ {x}$

## The Attempt at a Solution

So I did e^whole statement with ln(x/(x+1))*x, after that I multiplied that expression by 1/x/1/x, then I go ln(x/(x+1)/1/x, I tried taking derivative of top and bottom but it doesn't help with finding an answer.

Last edited:
Panphobia said:
So I did e^whole statement with ln(x/(x+1))*x, after that I multiplied that expression by 1/x/1/x, then I go ln(x/(x+1)/1/x, I tried taking derivative of top and bottom but it doesn't help with finding an answer.

It did for me. Try it again, maybe.

Panphobia said:

## Homework Statement

$lim_{x -> \infty} \left( \frac{x}{x+1} \right) ^ {x}$

## The Attempt at a Solution

So I did e^whole statement with ln(x/(x+1))*x, after that I multiplied that expression by 1/x/1/x, then I go ln(x/(x+1)/1/x, I tried taking derivative of top and bottom but it doesn't help with finding an answer.

I wouldn't e^ln(statement).I would write it as

##y= \displaystyle \lim_{x \to \infty} \left( \frac{x}{x+1} \right)^x##
##\ln{y}= \displaystyle \lim_{x \to \infty} \ln{\left[\left( \frac{x}{x+1} \right)^x\right]}##

Then you eventually solve for y. What do you get when you perform l'Hospital's rule? This is the correct procedure so far.

## 1. What is L'Hopital's rule?

L'Hopital's rule is a mathematical rule used to evaluate indeterminate forms in calculus. It states that if the limit of a function f(x) divided by g(x) as x approaches a certain value is an indeterminate form (such as 0/0 or ∞/∞), then the limit of the ratio of the derivatives of f(x) and g(x) is equal to the original limit.

## 2. What are indeterminate forms?

Indeterminate forms are mathematical expressions that cannot be evaluated using standard algebraic techniques. These forms often involve the division of one quantity by another that approaches 0 or infinity, leading to an undefined result. Examples of indeterminate forms include 0/0, ∞/∞, and 0*∞.

## 3. When should L'Hopital's rule be used?

L'Hopital's rule should only be used when the limit of a function divided by another function results in an indeterminate form. It is important to first simplify the function as much as possible before applying the rule. Additionally, L'Hopital's rule should only be used when the functions involved are differentiable.

## 4. Does L'Hopital's rule always work?

No, L'Hopital's rule does not always work. It is only applicable to specific types of indeterminate forms, and may not provide a valid solution in some cases. It is always important to check the conditions and assumptions of the rule before applying it.

## 5. Are there any alternatives to using L'Hopital's rule?

Yes, there are alternative methods for evaluating indeterminate forms such as using algebraic manipulation, substitution, or other calculus techniques. These methods may be more suitable depending on the specific problem and can provide a more accurate solution in some cases.

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