# Solving an indeterminate 0/0 limit without L'Hospital

• Hernaner28
In summary, the conversation discusses how to solve a limit without using L'Hospital's rule and whether or not the limit exists. The limit in question is x^(1/(x-1)) as x approaches 1. One person suggests using L'Hospital's rule indirectly, but another individual is unsure of a simple way to solve it without using the rule. The conversation also briefly touches on the types of limits that L'Hospital's rule can be applied to.
Hernaner28
Hi! I'd like to know how to solve the following limit without having to use L'Hospital's rule:

$$\displaystyle\lim_{x \to{1}}{x^{\frac{1}{x-1}}}$$

I've already solved it using the derivative rules but I want to know how to solve it without them. Thanks for your help!

EDIT: I was mistaken, it's not a 0/0 indetermiante limit, it's 1^infinity. Edit the title, thanks!

Does the limit from the left equal the limit from the right? Also, l'Hospital's rule doesn't apply to this kind of limit. What kinds does[\I] it apply to?

There's one point when you're solving this limit that you get an indeterminate 0/0 limit:

$$e\displaystyle\lim_{x \to{1}}{\frac{log(x)}{x-1}}$$

When you get there you may apply L'Hospital's rule but I want to know how to continue without using it.

Well, scurty, you can use L'Hopital, indirectly: If $y= x^{1/(x-1)}$ then $ln(y)= (ln(x))/(x- 1)$which is of the form 0/0. Unfortunately, Hernaner28, I don't see any simple way to do this other than L'Hopital's rule.

HallsofIvy said:
Well, scurty, you can use L'Hopital, indirectly: If $y= x^{1/(x-1)}$ then $ln(y)= (ln(x))/(x- 1)$which is of the form 0/0. Unfortunately, Hernaner28, I don't see any simple way to do this other than L'Hopital's rule.

I'm confused, I thought the limit doesn't exist? The limit from the left doesn't equal the limit from the right.

Edit: I'm on a mobile phone and I completely missed the x being raised to the power, I just saw the power. The x blended in with the lim. Okay, I understand what is being asked now.

## What is an indeterminate 0/0 limit?

An indeterminate 0/0 limit is a mathematical expression that results in a division by zero when evaluating a limit. This means that the value of the expression cannot be determined using standard algebraic methods.

## Why do we need to solve indeterminate 0/0 limits?

Indeterminate 0/0 limits often arise in calculus problems and are important for understanding the behavior of functions at certain points. They can also help us calculate derivatives and integrals, which have many real-world applications.

## What is L'Hospital's rule?

L'Hospital's rule is a mathematical theorem that provides a method for evaluating indeterminate 0/0 limits. It states that if the limit of the ratio of two functions is indeterminate, then the limit of the ratio of their derivatives will give the same result.

## Can indeterminate 0/0 limits be solved without using L'Hospital's rule?

Yes, there are other methods for solving indeterminate 0/0 limits. These include factoring, simplifying, and using other limit laws and theorems. However, L'Hospital's rule is often the most efficient and straightforward method.

## Are there any limitations to using L'Hospital's rule?

Yes, L'Hospital's rule can only be used when the limit is truly indeterminate, meaning that both the numerator and denominator approach 0. It also cannot be used for certain types of functions, such as those with discontinuities or infinite limits. In these cases, other methods must be used to solve the limit.

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