Deathfish said:Homework Statement
Find limit of (sin x - x)/(x - tan x) as x approaches zero
Homework Equations
Type 0/0 , use L'Hopital's rule, differentiate.
The Attempt at a Solution
Every time I apply the rule it gets more complicated
(cos x - 1)/(1 - sec^2 x)
(sin x)/(2 sec^2 x tan x)
Deathfish said:Ok usually how do you decide when to stop differentiation and use alternative method?
L'Hopital's Rule is a mathematical technique used to evaluate limits involving indeterminate forms, such as 0/0 or infinity/infinity. It states that if the limit of a function f(x) as x approaches a certain value is equal to 0/0 or infinity/infinity, then the limit of the derivative of f(x) as x approaches the same value will have the same value.
L'Hopital's Rule is used to solve trigonometric limits by taking the derivative of the numerator and denominator separately, and then evaluating the limit again. This process is repeated until a non-indeterminate form is reached, or until it is determined that the limit does not exist.
The indeterminate form 0/0 means that both the numerator and denominator of a fraction approach 0 as the variable in the function approaches a certain value. This can be seen in the expression sin x - x / x - tan x, where both the numerator and denominator approach 0 as x approaches 0.
Yes, L'Hopital's Rule can be used to solve other types of limits, such as infinity/infinity or infinity/0. However, it can only be used in cases where the limit is in an indeterminate form.
Yes, there are limitations to using L'Hopital's Rule. It can only be applied to limits that have indeterminate forms, and it may not always work for more complex functions. It is also important to note that using L'Hopital's Rule does not guarantee that the limit will be solved, as it may still be indeterminate after multiple applications.