How to Find the Limit of a Function, Analytically

In this Article we will use an example of a Rational Function to demonstrate How to find the Limit of a Function, Analytically. There are basically three different ways to find the Limit of a Function. Graphically, Numerically and Analytically. Depending on the type of Function we are working with, each method has its advantages and disadvantages.

Things You'll Need

• Paper and
• Pencil.

Instructions

1. 

   Given the following Rational function as an example,
   f(x) = (2 - x)/(x^2 -4), we are going to find the limit, Analytically, of f(x), as x->2. Please Click on the Image for a better Understanding.

2. 

   We will first try directly substituting x = 2 into the Rational Function. That is, f(2) = (2 - 2)/(2^2 - 4). the result is
   f(2) = 0/0, which is meaningless, in more mathematical terms, 0/0 is defined as, or called, an Indeterminate Number.

3. 

   Graphically, the graph of f(x) has a hole in it, at the point x = 2 in
   the Domain of f(x). To solve this problem we are going to find a function g(x), that is Exactly or Identically the same function as
   f(x), over the same Domain of f(x), except at the point x = 2. Please Click on the Image to see the Graphs of f(x) and g(x).

4. 

   We can actually simplify f(x) by factoring the denominator, (x^2 - 4). That is x^2 - 4 = (x - 2)(x + 2). So we can now rewrite f(x) as
   f(x) = (2-x)/[(x-2)(x+2)]. Since The expression 'x approaches 2' does not mean x = 2, we can divide both (2-x) and (x-2) by (x-2), since
   (x-2) is not equal to Zero ( 0 ). That is (2-x)/(x-2) = -1.
   SO f(x)= (2-x)/[(x-2)(x+2)] = -1/(x+2). We now call this function g(x),
   g(x)= -1/(x+2). The Limit of g(x), as x->2 is, by direct substitution,
   -1/(2+2) = -1/4. This limit is also the limit of f(x) as x->2. So the
   limit of f(x), as x->2, is -1/4. Please Click on the Image for a better understanding.