How to Find the Limit of a Rational Function with Numerator that has Fractions

This Article will use an example to find the limit of a complex fraction. Many students have a difficult time working with complex fractions. This article will demonstrate that it is not as difficult as it seems.

Things You'll Need

• Paper
• and Pencil

Instructions

1. 

   The function that we will be finding the limit of in this example is the Limit, as x -> 4, of [(x/(x+1)-(4/5)]/(x-4). The first thing we must do, when solving a limit function, is test by Direct Substitution. To do this, replace x, in the function, with the number that x is approaching. If this yeilds a real number, then that number is the limit. If this results in 0/0, then there is more work to be done. The function [(x/(x+1)-(4/5)]/(x-4), when x is replaced with 4, is as follows. [(4/(4+1)-(4/5)]/(4-4) = [(4/5)-(4/5)]/0 = 0/0. Please click on the image for a better understanding.

2. 

   If Direct Substitution results in 0/0, then the next thing we need to do is to rewrite the function. The rewritten function will be exactly the same as the original EXCEPT at the number that x is approaching. Since this function has fractions in the numerator, we need subtract both fractions. The function this creates is [(5x-4(x+1))/[5(x+1)]]/(x-4). Please click on the image for a better

3. 

   Next, we apply the order of opperations to the numerator, which will distribute the 4 to the expression inside the parentheses, and then combine like terms. This results in [(x-4)/[5(x+1)]]/(x-4). Now, we multiply the numerator of the whole fraction by 1/(x-4), in order to make the function a simple fraction. Since we have (x-4) as the numerator of one fraction, and (x-4) as the denominator of the other fraction, we can divide (x-4) out, making the result 1. This creates the function 1/[5(x+4)]. Please click on the image for a better understanding.

4. 

   Now that we have a new function, identical to the original function except at the point x=4, we can retry Direct Substitution. This should now give us a real number. When we replace x with 4, we get 1/[5(4+1)] = 1/[5(5)] = 1/25. This is the final answer to the limit. Please click on the image for a better understanding.