The limit of a function at a point a in its domain is the value it approaches as its argument approaches a. Intutively and informally, one could say that the limit of a function of a variable x at a is L if the function gets closer and closer to L as x gets closer to a.
For more on limits please read Paul’s Online Notes on the subject.
The concept of limit is fundamental to calculus and serves as the building block to define derivatives and definite integrals.
Limit of an expression involving multiple functions
The limit of an expression involving multiple functions can be obtained by taking the limit of the individual functions and then combining them to get the limit of the expression. Here is a simple example:
In some cases though this trick leads to unresolvable expressions. Common among such unresolvable expressions are limits of functions
leading to ratios of
This is when both L1 and L2 in our example above are either 0 or at the same time. Limits of such expressions cannot be determined only by the limits of the functions in the expression. Such an expression is in an indeterminate form.
Examples of indeterminate forms
where k is a real number and
leads to an expression of form .
creates an expression of form
The 7 indeterminate forms
There are 7 different indeterminate forms. and are the most common out of these 7. Other indeterminate forms are , , , , and .
Note: is not an indeterminate form. If and , then is of the form . If you however rewrite this equation in the form it is easy to see that both the multiplicands are tending to . is and hence the limit of this expression is .
For forms of the type , one could use the definition of a derivative to resolve the situation. By defintion, derivative is
which is of the form .
L’Hospital’s rule tells us that if we have an expression of the indeterminate form or then we must differentiate the numerator and differentiate the denominator separately and then take the limit.
In this case we get:
As an example,
L’Hospital’s rule can also be applied to indeterminate expressions of the form . In fact, such an expression can easily be converted to or form.
is then of the form .
A simple transformation to converts it to form.
Solving indeterminate forms
L’Hospital’s rule is the best method to solve indeterminate forms. In some cases an initial transformation allows an expression to be converted to or form after which the L’Hospital’s rule can be applied to the expression.
In some other cases, transformation could lead to forms where one can easily calculate the limits. Here is an example:
There are 7 different types of indeterminate forms. In order to find the limits of expressions in indeterminate forms, transform them to a or form and apply the L’Hospital’s rule.