next we will discuss how to find the limits of reciprocal functions using the analytic approach first let's figure out what does a reciprocal function look like reciprocal functions are the simplest rational functions and all reciprocal functions have this form a non-zero number c divided by x raised to some natural power n the number c is called the coefficient and the power of x is the degree of the denominator here are some examples of reciprocal functions note that the coefficient can be either positive or negative and the degree can be either even more odd when the coefficient is negative we frequently separate the negative sign from the coefficient and apply it to the entire fraction what makes reciprocal functions special is that all of them have the same end behavior and only four possible behaviors around zero depending on the sign of the coefficient and whether the degree is even or odd you can follow the link in the description of the video to figure out the four patterns for yourself but here's the summary of them what does that mean to us in terms of finding the limits the behavior of a reciprocal function with a positive coefficient and odd degree in the denominator can be described with the following limits f approaches to negative infinity as x approaches to zero from the left f approaches to positive infinity as x approaches to zero from the right for any point other than zero f approaches to its output at that point because reciprocal functions are continuous everywhere except zero and f approaches to zero as x approaches to positive or negative infinity the behavior of a reciprocal function with a positive coefficient and even degree in the denominator can be described with the following limits f approaches to positive infinity as x approaches to zero from the left f approaches to positive infinity as x approaches to zero from the right for any point other than zero f approaches to its output at that point because reciprocal functions are continuous everywhere except zero and f approaches to zero as x approaches to positive or negative infinity the behavior of a reciprocal function with a negative coefficient and odd degree in the denominator can be described with the following limits f approaches to positive infinity as x approaches to zero from the left f approaches to negative infinity as x approaches to zero from the right for any point other than zero f approaches to its output at that point because reciprocal functions are continuous everywhere except zero and f approaches to zero as x approaches to positive or negative infinity finally the behavior of a reciprocal function with a positive coefficient and odd degree in the denominator can be described with the following limits f approaches to negative infinity as x approaches to zero from the left f approaches to negative infinity as x approaches to zero from the right for any point other than zero f approaches to its output at that point because reciprocal functions are continuous everywhere except zero and f approaches to zero as x approaches to positive or negative infinity as a result at any given moment we should be able to associate any reciprocal function with the appropriate limit statement based only on the coefficient and the degree of the denominator as the following chart summarizes instead of memorizing the chart try to understand the behavior of the graph for different combinations of degrees and coefficients and then sketch a quick draft of the graph to assist you with finding the limit we discussed how to find a limit of a reciprocal function using the analytic approach