Consider 2 functions, and . Let be a value at which these functions are defined.
I) If and , then the limit takes the form .
II) If , then takes the form
III) If and , then the limit takes the form
IV) If , and , then the limit takes the form .
V) If , then the limit takes the form .
VI) If and and , then the limit takes the form
VII) If and and , then the limit takes the form .
These are known as the indeterminate forms. The limits are evaluated either by L’Hosptial’s rule or by substituting an equivalent infinitesimal.
L’Hosptial’s Rule (French : )
The rule can be proved using Taylor’s theorem. It says, if and are at or and , then
This rule is sometimes applied on th derivatives, if all derivatives of lesser orders are .
This is used for evaluation of form. One of the functions can be replaced by another, if they both converge to at a point and the limit of their ratio at that point is . For example,
Hint for MCQs
One can try substituting a value of closer to (but not equal to) the actual limit. Evaluate the function using the calculator. The answer will be closer to the actual limit. We’ve actually used the concept of limit here.
Explanation: Consider the limit
This is of the form . Let’s put in the function .
We get as .
On substituting , a value closer to , we get .
On substituting , a value closer to and , we get . Clearly, the limit is .