How to Find The Limit of a Function, Numerically

The function used as the example for this article has the expression x squared in it. Since this is difficult to express correctly in the text portion of the steps below, the steps will be mostly shown in the images.

Things You'll Need

• Paper and
• Pencil

Instructions

1. 

   The expression, 'as x approaches some number c', written in notation form; 'x->c', means as x approaches c on the Number Line, from the left,and from the right simultaneously. The first step in trying to find the Limit of a function,as x->c, is to try Substituting c directly for x in the given function. That is replace 'x' with the number c, that x is approaching. If this yeilds a real number, including zero, then that Number is the Limit of the Function. If the result is a number being divided by zero, #/0, then the Limit does not exist. If the result is, 0/0, then we have to find a another Function that is exactly the same as the original Function for all x's in that Domain except for the point, x=c. Please click on the image to see the function, and to get a better understanding.

2. 

   The method we are going to use to find the Limit of the Function given in the example, is the Numerical method. We are going to approach the number, c=2, from the Left, with numbers that are extremely close to the number c=2. First, we use numbers that are slightly less than 2. We will use 1, 1.5, 1.7, and 1.9. When these numbers are substituted into the Function, the results are, respectively, -1/3, -1/3.5, -1/3.7, and -1/3.9. Please click on the image for a better understanding.

3. 

   Next, we will approach, c=2, from the right, that is, we will use numbers that are slightly larger than 2. We will use 3, 2.5, 2.3, and 2.1. When these numbers are substituted into the function, the results are, respectively, -1/5, -1/4.5, -1/4.3, and -1/4.1.

4. 

   The next step is to enter all of the above numbers and their corresponding results into a chart. For this particular example The chart should have two rows,and 10 columns. We will label The first column, 'x' and 'f(x)'. The top row will have the numbers that are approaching 2, as well as the number 2. These should be in numerical order. The bottom row should have all the corresponding results for each respective number. Please click on the image for a better understanding.

5. 

   Now, we need to find out what number f(x) is approaching, as x is approaching 2. We take the two sequence of numbers, on either side of x=2, and see what number they are approaching, from the two different directions. We can see that, That number is -1/4. This is the Limit of the function.