The rule can be used to evaluate limits that are of the form .
Consider two functions and which are differentiable in the neighbourhood of the point (except possibly at the point itself). Let in this neighbourhood.
The rule says that
provided that the limit exists. Although the rule is applicable to limits of the form , you should be able to understand that other indeterminate forms like can be reduced to these two indeterminate forms using appropriate algebtain manipulations.
You are urged to think of some (non-rigorous) justification for this rule.
Lets apply this rule on some examples.
We have encountered both these limits in the unit on Limits. Here, we’ll re-evaluate them using the rule.
This is what we got earlier
This limit is of the indeterminate form . Lets first convert it into the form .
The limit is of the indeterminate form , so we apply the rule:
Now, we know that does not exist since is an oscillating function and does not converge to any particular value. What does this imply for our current limit? Does it not exist ?
Think about the expression carefully:
Thus, a limit does infact exists while the rule says that it does not exist. Why?
This is because the rule is not applicable here. Go back to the definition of the rule which says that if the latter limit exists.
In this example, you cannot apply the rule on the expression in () since the limit for the expression obtained after differentiation (the one in ()) does not exist.