L'Hopital's Rules: Solving Indeterminate Forms in Class 12 Math Notes Reveal"

Hi Everyone! Today in this video i am going to reveal notes of class 12 math " application of derivative indeterminate form ( L hospital's rules) .
I'd be happy to help!
*What is the Indeterminate Form?*
In calculus, the indeterminate form occurs when we try to evaluate a limit of the form:
0/0 or ∞/∞
These forms are called "indeterminate" because we can't immediately determine the value of the limit. It's like trying to divide by zero - it just doesn't work!
*What is L'Hôpital's Rule?*
L'Hôpital's rule is a powerful technique for evaluating limits that are in the indeterminate form. It's named after the French mathematician Guillaume François Antoine, Marquis de L'Hôpital.
The rule states that if we have a limit of the form:
lim (f(x) / g(x)) = 0/0 or ∞/∞
where f(x) and g(x) are both functions of x, and:
f(x) = g(x) = 0 (or f(x) = g(x) = ∞)
then we can evaluate the limit by replacing f(x) and g(x) with their derivatives, and taking the limit again.
*The Formula:*
The formula for L'Hôpital's rule is:
lim (f(x) / g(x)) = lim (f'(x) / g'(x))
where f'(x) and g'(x) are the derivatives of f(x) and g(x), respectively.
*How to Apply L'Hôpital's Rule:*
Here's a step-by-step guide to applying L'Hôpital's rule:
1. Identify the indeterminate form: 0/0 or ∞/∞
2. Check if both functions are equal to zero (or infinity): f(x) = g(x) = 0 (or f(x) = g(x) = ∞)
3. If true, replace f(x) and g(x) with their derivatives: f'(x) and g'(x)
4. Take the limit again: lim (f'(x) / g'(x))
*Example:*
Let's say we want to evaluate the limit:
lim (x -is grater than 0) (sin(x) / x)
This is an indeterminate form because it's of the form 0/0.
To apply L'Hôpital's rule, we need to find the derivatives of sin(x) and x:
d/dx (sin(x)) = cos(x)
d/dx (x) = 1
Now, we replace sin(x) and x with their derivatives:
lim (cos(x) / 1)
Evaluating this limit is easy! We get:
lim (cos(x)) = 1
So, the original limit evaluates to:
lim (sin(x) / x) = 1
That's it!
*Key Points:*
* L'Hôpital's rule helps us evaluate limits that are in the indeterminate form 0/0 or ∞/∞
* We replace f(x) and g(x) with their derivatives, and take the limit again
* It's a powerful technique for solving limits that seem impossible to evaluate
I hope this explanation helps you understand L'Hôpital's rule in simple terms!
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L'Hopital's Rules Solving Indeterminate Forms in Class 12 Math notes reveal"