# How to solve limits with fractions in the numerator

## How to Solve Limits with Basic Algebra Techniques for Finding Limits of Rational Functions

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In the previous section we saw that there is a large class of functions that allows us to use. The purpose of this section is to develop techniques for dealing with some of these limits that will not allow us to just use this fact. The first thing that we should always do when evaluating limits is to simplify the function as much as possible. In this case that means factoring both the numerator and denominator. Doing this gives,. Therefore, the limit is,.

By limits at infinity we mean one of the following two limits. The proof of this is nearly identical to the proof of the original set of facts with only minor modifications to handle the change in the limit and so is left to you. The first part of this fact should make sense if you think about it. So, we have a constant divided by an increasingly large number and so the result will be increasingly small. Or, in the limit we will get zero.

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What happens to the value of a function as x becomes infinitely large? Does the function also become infinitely large? Does it reach a maximum or minimum? What in fact happens? We know that we cannot divide by zero but what happens if we divide by a number very near to zero? We will see that sometimes the graph of the function goes up or down almost perpendicularly near this pont and sometimes the graph cannot go through the point leaving an infinitisimal hole.

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Hint: how about simplifying this by multiplying numerator and denominator by 4x? Or try to simplify 14(4+x)+1x(4+x)?.
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