**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 , in that case we can say that f(x)/g(x) is an indeterminate for of the type at x = a.

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 , then it is equal to the limit of f(x)/g(x).

**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 exist or (∞ , -∞), then

**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 has a finite limit, or if it is +∞ or-∞ , then

**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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