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.
Panphobia said:Homework Statement
[itex]lim_{x -> \infty} \left( \frac{x}{x+1} \right) ^ {x}[/itex]
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.
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.
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*∞.
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.
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.
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.