Overview
The L’Hospital’s rule rule was given by Guillaume de L’Hôpital. The technique discussed in this method is used to evaluate indeterminate
Suppose we have two functions f(x) and g(x) and both are zero at x = a, then the fraction f(a)/ g(a) is called the indeterminate form 0/0.
In other words
Let we have two functions f(x) and g(x) and-
Then
Is an expression of the form 0/0
Now, Let we have two functions f(x) and g(x) and-
Then
Is an expression of the form 
Some other indeterminate forms are
In case we get indeterminate form we apply L’Hospital’s rule.
L’ Hospital’s rule for 0/0 form-
Working steps-
1. Check that the limits f(x)/g(x) is an indeterminate form of type 0/0.
(Note- we can not apply L’Hospital rule if it is not in indeterminate form)
2. Differentiate f and g separately.
3. Find the limits of the derivatives .if the limit is finite ,
Example-1: Evaluate
Sol. Here we notice that it is an indeterminate form of .0/0
So that , we can apply L’Hospital rule-
Note- Suppose we get an indeterminate form even after finding first derivative, then in that case , we use the other form of L’Hospital’s rule.
If we have f(x) and g(x) are two functions such that
If 
Example-3: Evaluate
Sol. Let
then
And
But if we use L’Hospital rule again, then we get-
L’Hospital’s rule for ∞/∞ form
Let f and g are two differentiable functions on an open interval containing x = a, except possibly at x = a and that
If 
Theorem- If we have f(x) and g(x) are two functions such that
If exist or (∞ , –∞), then
Example-5: Find , n>0.
Sol. Let f(x) = log x and g(x) = x^n
These two functions satisfied the theorem that we have discussed above-
So that,
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