# Indeterminate Forms

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 :

1. 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.
2. 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.
3. 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.
4. 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.
5. 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.

Example-1:
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

Example-2:
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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