If a function is know to be continuous at , then by the definition of continuity we have, .
We will take is as known that the elementary functions – polynomial, rational (polynomial/polynomial), trigonometrical, exponential, logarithmic) – are continuous everywhere.
Therefore, although not very exciting, we are able to say such things as:
A very useful rule (it’s a theorem really) for finding limits is L’Hôpitals rule (LH), which is really a collection of rules. The most comprehensive form of the rule is as follows:
Suppose you wish to find something like where [ ] may be .
Suppose further that where either
– ( type), or
– ( type).
Then the LH rule says look at and if you can do it, say find it to be ( ) then the original limit is the same.
Danger: Do not use LH when it is not applicable.
For example is not the same as .
The LH rule gives us with no effort some useful results. For example: