Assume a function which is undefined at x=a but it may approach a limit as x approaches a. The process of determining such a limit is known as evaluation of indeterminate forms. The L’ Hospital Rule helps in the evaluation of indeterminate forms. According to this rule
Provided that both f’(x) and g’(x) exist at x = a and g’(x) ≠ 0.
Types of indeterminate forms :

Type
Suppose f(x) = 0 = g(x) as x→ a or as x→ 0
This form can be solved directly by the application of L’ Hospital rule.
Provided that both f’(x) and g’(x) exist at x = a and g’(x) ≠ 0. 
Type
Suppose f(x) = ∞ = g(x) as x→ a or as x→ ±∞. This form can be solved by first converting it to the type as
Now we can apply L’ Hospital rule as usual to solve it. It is advised to convert to 0/0 form as the differentiation of numerator and denominator may never terminate in some problems. 
Type
Suppose f(x) = 0 and g(x) = ∞ as x→ a or as x→ ±∞ then the product f(a).g(a) is undefined. We need to solve it by converting it to the type 0/0 or ∞/∞.
or
Now we need to apply L’ Hospital rule. 
Type
Suppose f(x) = ∞ = g(x) as x→ a. this type is solved by again converting to the 0/0 form by following method :
As we achieve 0/0 form, now we can apply L’ Hospital rule. 
Type
To evaluate these forms consider:
Taking logarithm both sides
Taking the limit as x→ a or x→ ±∞
Then
Note –
If f’(x) and g’(x) do not exist at x=a then we need to perform the differentiation again until the derivatives of f(x) and g(x) become valid.
Example1:
Evaluate
Explanation :
As the given function assumes 0/0 form at x = 1, so we can directly apply L’ Hospital rule.
This forms 0/0 form again. Hence we apply L’ Hospital rule again.
and
Thus
Example2:
Evaluate
Explanation :
The given function assumes 0.∞ form. We will first rewrite it in form.
Now we apply L’ Hospital rule to get
This forms form again. We rewrite it in 0/0 form as
Now apply L’ Hospital rule again.
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