# What is the Addition Rule for Limits? - Definition & Overview

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this lesson, you will learn how to identify problems in which you can apply the addition rule for limits. You will also learn in what instances you cannot use this very helpful rule.

## Definition

Recall that the limit is the value that the function gets close to as you get closer to a particular point. In math symbols, defining the limit looks like this.

What this says is that when you take the limit of f(x) as it approaches c, it will either approach or become the value L.

When you need to take the limit of more complicated functions, the addition rule comes into play. The addition rule helps you to find the limits of more complicated functions that are the sum of two or more smaller functions. The rule tells you that you can split up the larger function into the smaller functions and find the limit of each and add the limits together to get the answer.

This rule makes it easy to figure out limits where you want to take the limit of a larger function that is the addition of two simpler functions that you know the limits to.

Of course, the addition rule comes with its own general formula.

## Formula

The formula for the addition rule shows how to apply it in all situations where you have functions that are the sum of smaller functions.

Like with many things in math, though, this formula has a limitation or condition under which it will work. Outside of this condition, the formula won't work.

## Condition

What is this condition? It is that the limits of the smaller separated functions must exist. If they don't exist, then you can't use this rule because it won't help you.

For example, I want to take the limit of the following function.

The addition rule says that I can break this larger function into the limits of the smaller functions like this.

This is easier to work with because we know how to find the limits of the smaller functions themselves instead of the larger function as a whole.

## Applying the Addition Rule

Now, let's see how to apply this rule to limits problems.

Let's say we are trying to find the limit of the following function.

The addition rule tells us we can split this function into two smaller functions before taking the limit.

Doing this we see that the smaller functions are easy to find the limits for. We see that they are polynomials and so we can plug our x limit directly into the smaller functions to find their limit.

After doing that, we see that our answer for this limit is 4.

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